Maintaining validity of ZKP throughout re-randomization of homomorphic ciphertext without linkability to previous ciphertext
First (if it's not the case) you have to read carefully the original Groth-Sahai paper : https://eprint.iacr.org/2007/155.pdf

We can focus on a concrete example to understand why it's possible: in the "SXDH" setting for example (page 24);
Let suppose we have a commit $\vec{c_1}$ for a vector of elements of $G_1$ and
a commit $\vec{c_2}$ for a vector of elements of $G_2$. And a proof $(\vec \pi, \vec \theta) $

You can change randomize the commitment $c_2$ (by adding $R'\vec{u}$ to $\vec{c}$ for example), but if you do this the proof will be no more valid.

Then you have to add also the corresponding randomness to the proof, here it means you have to add to $\vec\pi$ $R'\Gamma c_2$.

More generally you have to randomize the commitment and the proof in the same time (Notice that in the groth sahai model commits are not considered as cipher but as part of the proof).