Mass balances some momentum and some energy into a location via $E^2-(|\vec p|c)^2=(mc^2)^2.$
So basically, every time you created some mass $m$ you had to put some energy $E$ into that location and some momentum $\vec p$ into that location and then it proceeds to take off with velocity $\vec v=\vec p c^2/E.$
So its the energy and the momentum that affect the curvature of spacetime. The mass is just a balance between the energy and momentum that are tied together in a way that moves together with the velocity $\vec v=\vec p c^2/E.$
The mass itself has absolutely no effect on the spacetime whatsoever except for the momentum and the energy associated with the mass.
So the energy and momentum flowed in, they may or may not be converted into mass by coupling together. But the curvature doesn't change unless the motion of the energy and momentum changes.
For instance if you had some energy flowing in from two directions towards the same place they might rush in fast because they have a small mass. But as they collide the momentums could cancel (they point in opposite directions) and form a larger mass collection that then moves slower.
The curvature doesn't notice anything until the energy and momentum starts flowing slower. The actual conversion into more mass had no direct effect at all.