"On their back" or "on their backs"?

It is decidedly the Singular Back. This is because it is the therapy that is being evaluated but the results are singular. The children are not grouped or lined up as in a race to see who can move. The plural children is only used to describe the percentage of those who are able to move. Each child moves only on his or her own back, no other, Singular.

The semantic, grammatical, and logical arguments clearly suggest it's 'correct' to use the plural, and that's what most people do.

However, despite the fact that I doubt if any style guide endorses the singular, it seems that about 10% of usages for back persist in using the singular.

For reasons which escape me, that 'incorrect' minority rises to nearly 25% when the body part in question is chest. In both cases the evidence strongly suggests the incorrect usage is becoming more widespread.

Personally I believe it's a situation where grammarians backed the wrong horse, and their blind prescriptivism will eventually be defeated. People quite naturally want to use the singular when the number of [body parts, whatever] is immaterial, and only the plurality of [babies, people] is relevant. Increasingly, it seems, they're prepared to do this even at the risk of being considered illiterate.

Plural, because we are, in fact, dealing with eight little backs. And a back is a back, no matter how small. A child has a back, but children have backs.

I'm sure others will back me on this.