Let's start by assuming that a transversible wormhole can be constructed in the most "likely" way we can currently imagine... Say, we created two entangled black holes and use "negative matter" to make them transversible. Say they also have charge so you can separate them with a magnetic field and move the "mouths" around. Say you use a gas-giant worth of mass so that they don't evaporate in your face.
Now, I do completely realize some of it is very hypothetical and probably far from from being possible. But let's treat it as a thought experiment and assume all these things.
Now let's say we drive one "mouth" to an orbit around the Sun and the other to an orbit around Proxima Centauri (again, assume you got enough fuel to do that). In this situation, the Sun would "see" Proxima Centauri through the wormhole and the other way around.
My question is - would the gravitational effects of the stars flow through the wormhole (perturbing the stars as a result) the same way that light can travel through?
I'm assuming spacetime has to still be smooth so it's not clear to me how the different relative curvatures will (or are supposed to) align after the wormholes are made?