I am wondering if there is an identity based proxy re-encryption scheme (such as the Green-Ateniese scheme https://eprint.iacr.org/2006/473.pdf) that converts an encryption computed with ID1, system parameters1 into one intended for ID1, system parameters 2, i.e., the ID is the same but the system parameters change.
This concept is called inter-domain identity-based proxy re-encryption. In this case, the notion of "domain" implies a separate KGC and a set of system parameters. There are a couple of schemes doing this:
Tang, Q., Hartel, P., & Jonker, W. (2008, December). Inter-domain identity-based proxy re-encryption. In Information Security and Cryptology (pp. 332-347). Springer Berlin Heidelberg. Han, J.,
Susilo, W., & Mu, Y. (2013). Identity-based data storage in cloud computing. Future Generation Computer Systems, 29(3), 673-681.