States can NOT garnish VA Disability Compensation for back child support. Or at least Missouri can NOT do so. Remember that according federal law , VA Disability Comp is NOT considered income. AS others have mentioned, only the federal government can garnish VA Disability Comp.
The state of Missouri tried garnishing my VA Dis. Comp when I fell behind on child support while waiting for Social Security Disability to kick in and also waiting for an increase in my VA Disability rating. Try as they might, the state could not touch my Disability Comp. So check with your state laws. But I am pretty darn sure that states can NOT touch VA Disability Comp since it is not legally income per federal law. Missouri definitely can't.
Now with that said, I do not know if federal law have changed since 2007 when I last had to deal with child support payments. One my exe started getting money from Social Security for both children, my child support stopped all together. In fact I actually got a refund from the state after they found out my exe was getting money from Social Security while still getting paid child support.
As far as gettin extra each moth for dependents, the VA does not care if the children reside full or part time with the veteran. They are still the veteran's dependents no matter what. I still got paid extra each month for both of my children from the VA even though their mom had primary custody. And that did NOT change even when my exe started getting Social Security benefits for the children either.
Side note: Social Security benefits goes to the parent that the child primarily lives with and also has legal custody. I know this since my exe was getting money from Social Security for both kids until the oldest moved in with me permanently. She go money for the younger son while I got money for the older son.