Randomizable zero-knowledge proofs?
Yes there are. I guess you're looking into non-interactive zero-knowledge (NIZK) proofs, since you're assuming that you have at hand a proof $\pi$. A common approach for building NIZKs in crypto is to rely on the Groth-Sahai methodology, which produces highly structured NIZKs using elliptic curves equipped with a bilinear maps. It was first observed in this paper that Groth-Sahai proofs can be randomized - i.e.,  you can randomize a statement together with its proof so that the randomized proof is indistinguishable from an honest proof computed on the randomized statement. This was used to build anonymous credentials with strong properties, and had many other applications in subsequent works.