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There's a well developed theory of deformations of formal groups. You can read a little bit about some aspects of it in the following (excellent) answer on MO mathoverflow.net/a/114695/22810
I find the cases of characteristic sheaves on open/closed balls very instructive. Try to plug in such a sheaf into the machine and see what you get at the end. After that you can try more general open/closed (perhaps some mildly singular) submanifolds. The general constructible sheaf always has filtrations whose quotients look (at least locally) like these so once you understand these examples you should be in good shape.
