If the spacetime dough disappears, so do the raisins
In their answer, the galaxies will begin to fall towards one another, yes,
but only if they're both still
under the influence of their mutual gravities. If that mutual attraction is
less than the Hubble flow for the distance between them, both will eventually get
caught up in the flow and
start to recede,
despite of initially moving together (or not at all, if not under their mutual gravitational influence i.e. too far away). It's only when two or more galaxies are under the influence of their mutual gravitational fields, like the Milky Way and Andromeda, that the local Hubble flow becomes negligible in comparison to their gravitational connection and as far as the flow is concerned they're considered to be one object.