Medical Research
A number of medical researchers have written about and presented evidence on the healthy aspects of barefoot activity for children (and also adults). The collection of references at Bare Feet in Medicine is quite extensive.  The book, Take Off Your Shoes and Walk by Simon J. Wikler is particularly good. In this book, podiatrist Dr. Wikler advocated going barefoot for better foot health (especially for children). He also explained how shoes often deform and distort our feet. An overview of the book is here. Also see Growth and Development.

Lynn Staheli, MD
Quoted in The New York Times on  Aug 14, 1991

Children with the healthiest and most supple feet are those who habitually go barefoot, according to Dr. Lynn T. Staheli and a growing number of other pediatric orthopedists. His studies of developing nations show that non-shoe-wearers have better flexibility and mobility, stronger feet, fewer deformities, and less complaints than those who wear shoes regularly. He says that, when a child must wear a shoe, it should be lightweight, flexible, shaped more or less quadrangularly, and above all, should not have the arch supports and stiff sides once deemed necessary to give the foot support. Many pediatric orthopedists strongly oppose "corrective" or "orthopedic" shoes for straightening foot and leg deformities like flat feet, pigeon toes, knock-knees, or bowlegs. Dr. Staheli and others contend that there is no evidence that corrective shoes correct anything, and that most of the supposed deformities correct themselves in almost all cases.  (click here for the full version of this article).

Dr. Staheli is a very prominent pediatric orthopedist ...

Publication:  Staheli, Lynn T. Fundamentals of pediatric orthopedics. Raven, 1992. WS270 S781f 1992
Award:  Lynn Staheli, MD, received the Distinguished Service Lectureship Award, Orthopedic Section, American Academy of Pediatrics
Dr. Staheli  is on the Faculty of the University of Washington Department of Orthopædics in the Pediatric Orthopædics section. In 1992 ... he was listed as Director of Orthopedics at the Children's Orthopedic Hospital and Medical Center in Seattle

Dr. Paul W. Brand

Barbara Platte interviewed Dr. Brand in 1976. Here is the account from the San Francisco Chronicle

What every kid seems to know instinctively -- that going barefoot is good for you -- has been confirmed by an orthopedist and rehabilitation specialist who has studied foot problems in various parts of the world.
"A high proportion of the world's population walks barefoot most of the time, and the average person who walks barefoot has much healthier feet than the average person who wears shoes, says Dr. Paul W. Brand, chief of the rehabilitation branch of the U.S. Public Health Service Hospital, Carville, La. and professor of surgery a Louisiana State University Medical School.
Dr. Brand conducted an orthopedic clinic in India for 18 years and has also treated orthopedic problems in Ethiopia and the United Kingdom. On the basis of these experiences he suggests that maybe we in America should walk barefoot for at least part of each day.
Common foot problems in the United States include corns, bunions, hammer toes, athlete's foot and ingrown toenail - but none of these is a problem in countries in which most people go barefoot.
"Every one is a product of footwear. They are caused either by poorly designed or poorly fitted shoes, or in such conditions as athlete's foot, by the simple fact of wearing shoes that prevent free access of air between the toes," he says.
Shoe-wearers also receive mechanical stress from walking on the same part of the foot with each step, while the rough, uneven thrust of the ground is transmitted to a different part of the barefoot walker's foot with each step.
In shoes, Dr. Brand says, sensitivity, mobility, and intrinsic muscle strength of the feet are lost.
" The barefoot walker receives a continuous stream of information about the ground and about his own relationship to it, while a shod foot sleeps inside an unchanging environment. Sensations that are not used or listened to become decayed and atrophy. There is a sense of aliveness and joy which I experience walking barefoot that I never get in shoes," he says.
Such direct contact with the ground has a great deal to do with preventing fractures of the ankle from "turning the foot over" by stepping on a rock or edge of the pavement. In India Dr. Brand saw no ankle fractures except in those who wore shoes.
The reason for this, he says, is that a barefoot person has instant information about the situation relayed by the touch-sensitive nerves in the skin of the sole and gets the information in time to avoid putting his weight on the unstable foot.
The person wearing shoes may not get this information in time to remove his weight from the leg to prevent a fracture.
In the Olympic Games, Indian and Pakistani teams have won several gold medals in field hockey while playing barefoot, and have had an impressive injury-free record. They have now turned to shoes in international competition but only in self-defense against players who wear boots with cleats and studs, Dr. Brand notes.
Going barefoot seems to have an additional advantage in preventing arthritis of the hip. For persons of comparable age, this is seen more frequently in Westerners and Indians who wear shoes than in the barefoot population.

From The Pilbara Times, Australia (31 Jan 1980) ....

Care For Kids Then Care For Their Feet

Edited extracts from an interview with the president of the Australian Podiatry Association ........
"[Children's] bones are soft cartilage, easily compressible, and they don't feel pain until the damage is done," said the president. The comment was made on the need for parents to be aware of the damage that is being done by children wearing shoes. The Australian Podiatry Association provides a free community service in an effort to create greater awareness of the need to safeguard children's feet.
30% of the children checked in the associations survey (2,500 children) in an eastern Australian state were found to be wearing footwear inadequate for day long use.
"The effects of childhood foot damage can show up in posture and gait in the early twenties," the president said. "The inability for a person to stand for any length of time without stress can also be attributed to early foot problems. Shoes must take a lot of the blame for claw toes, under and over riding toes, bunions and ingrown toe nails, not to mention corns and callus'."
"Australian children probably have broader feet now than they did ten years ago, because so many go barefoot. Between the ages of 7 and 12 years, growth is fairly rapid. Don't hesitate to let them be barefoot. It won't spread or flatten normal healthy feet as the foot only grows as long as the ligaments allow anyway. The majority of foot damage is preventable if parents take proper care of their children's feet by allowing them to grow naturally - barefootedly."

Say OK to Bare Feet

Recent research has shown that children are more likely to have good posture if they go without shoes as often as is possible and safe, according to Janet Perry, M.P.T., a physical therapist with Rehabilitation Network in Portland, Oregon. "That way they get more 'sensory information' from their feet and will have better walking and postural skills." Children should be allowed to go shoeless ... in   places where it's safe to walk barefoot as a way to improve their posture, according to Perry.

Physician says children need shoes that fit,
or no shoes at all

Style and price are not nearly as important as fit when it comes to choosing a good children's shoe. In fact, the best shoe for a child is often no shoe at all, according to Dr. Carol Frey, associate clinical professor of orthopaedic surgery in Manhattan Beach, Calif..  "We are no different from any other animal. We don't need shoes for proper foot development," explains Frey. Walking is a collaborative effort requiring constant communication between the brain and feet. Nerve endings on the bottom of the feet sense the ground beneath and send signals to the brain that help it determine how and where weight should be distributed with each new step. Shoes alter that feedback to the brain. The thicker the sole, the more muffled the message. "Children are forced to walk with their feet further apart to keep their balance," Frey says. 
Frey has just compiled a list of shoe-buying tips for the American Orthopaedic Foot and Ankle Society. "Shoes are not necessary for support or development of the arch, they only protect the feet from the environment," said Fry, who routinely treats foot deformities caused by ill-fitting shoes. Babies and crawlers need only wear socks or booties to keep their feet warm, she said. Toddlers, too, should be allowed to go sans shoes whenever they are in a protected environment. Going barefoot helps children develop stronger and more coordinated foot muscles, Frey explained. Studies also show that barefoot children learn to walk more quickly and have fewer falls. 
Besides growing up around man-eating lions, there’s a good reason Kenyans have won the last 10 consecutive Boston Marathons: They rarely wear shoes. This makes their feet extremely strong and far less susceptible to disorders such as fallen arches, says Dr. Frey.

Many people believe that "arch support" is an  important contribution from shoes. A number of researchers and doctors are challenging this assumption. See these extracts from an article by Dr. William Rossi. Also see "Why Shoes Make 'Normal' Gait Impossible" by the same author.

For children,  arch-development issue has been borne out by a detailed study. An extract from this study follows.

Udaya Bhaskara Rao and Benjamin Joseph.
"The Influence of Footwear on the Prevalence of Flat Foot"
The Journal of Bone and Joint Surgery, 74B(4), 1992, pp. 525-527.

Abstract: We analyzed the static footprints of 2300 children between the ages of four and 13 years old to establish the influence of footwear on the prevalence of flat foot. The incidence among children who used footwear was 8.6% compared with 2.8% in those who did not (p < 0.001). Significant differences between the predominance in shod and unshod children were noted in all age groups, most marked in those with generalized ligament laxity. Flat foot was most common in children who wore closed-toe shoes, less common in those who wore sandals or slippers, and least in the unshod.

Our findings suggest that shoe-wearing in early childhood is detrimental to the development of a normal longitudinal arch.

In Europe and America flat foot is a common reason for attendance at a children's orthopaedic clinic, but in India children are seldom brought for treatment for flat foot. The few children who do attend with this complaint are from affluent urban families and they all wear shoes. In our clinic we have never seen a child from the farming community or from the family of a manual labourer who complained of flat foot.
.....
The high concentration of flat foot among six-year-old children who wore shoes as compared with those who did not, implies that the critical age for development of the arch is before six years.
.....
Our cross-sectional study suggests that shoe-wearing in early childhood is detrimental to the development of a normal or a high medial longitudinal arch. The susceptibility for flat foot among children who wear shoes is most evident if there is associated ligament laxity. We suggest that children should be encouraged to play unshod and that slippers and sandals are less harmful than closed-toe shoes.

...... This study also mentioned that the slipper and sandal-wearing children tended to play barefoot (since this footwear is more easily removed) and that this was believed to account for the intermediate flatfoot incidence rate.