The game progress/purchases are not updating for the Restricted Profile that my child uses on my device.

If you have a separate Restricted Profile set up for your child on your Android device, you may not see purchases made on your regular profile refelected in the child's profile.

To fix this, enable access for the "Google Games" app for the child account via the Restricted Profiles configuration.  This should allow the game to sync progress with the regular account.