It is easy to see (and presumably well-known) that the operation of attaching manifolds along a submanifold (as defined Characteristic Classes of a Fibered Sum) can be expressed as a push-out in the Smooth category, so long as the construction includes collars. The connected sum follows as a special case where the submanifold is a point. Furthermore, Thom's "shperical modifications" can also be treated as a connected sum. Basic facts about these operations (well-defined under isotopy of the embeddings, associative, etc.) can be proven as corollaries of theorems about push-outs.
I've done some work on my own developing this idea, but I'm wondering if there's some theory already in place that could help me save some time or point me in better directions. I haven't come across anything in my searches. What I'm looking for specifically are references which treat these operations as pushouts, and develop them as such. (What I am NOT looking for are modifications of the smooth category which make it closed under pushouts)