Say you have a black hole, and behind the black hole (from your perspective) is some mass, Alpha.
Mass Alpha has its own gravitational field. The field at some point passes "through" the black hole for us to observe.
Does the black holes gravity inside the event horizon affect our observation of mass Alphas gravitational effects? Why could the gravity pass through the event horizon? Is it additive? Or does the black hole "eat" some of the gravity? Or something else happens?

Additionally, if gravity propagates at a specific speed and is not instantaneous, shouldnt we observe a Doppler effect?
Why does a black holes gravity have to be infinite? Couldn't there be a maximum gravity in the same way that light has a definite speed?