I was wondering if their is a general strategy for computing path integrals with a mix of boundary and bulk integral actions. Do people use divergence theorem to convert the action into bulk integrals, or is there some other trick?
The usual Feynman rules may still be derived. If there are both boundary fields and bulk fields, one needs to include the bulk-to-boundary propagators, too. The same problem has to be solved in the AdS/CFT correspodence. Search for "Witten diagrams" to see how the Feynman diagrams are ultimately done in this context.