I was thinking only about various transformations and how they change the metric. In particular, is it possible to think of transformation changing not only the coordinate chart on a manifold but also defining a new manifold. In this respect, every "physical" transformations shouldn't have this ability(change manifolds by boosting ourselves) whereas others could maybe change the manifold. I am not very fluent in differential geometry but i believe this is more a mathematical question.
As for Poincare in GR, I would plug in the same as in SR... Or almost the same ;). It would give me GR metric in some boosted ref. of frame. For example, Schwarzschild metric is the one in which mass in the center is at rest and observers(frames) naturally move towards the center. In order to get myself hoovering over the BH at fixed distance i need to boost myself properly (it should even contain some accelerations) and this would be my Poincare transformation. You are probably right that not simple Poincare with constant velocity but some local version of it.
Hopefully someone else will join the discussion ;).