Proxy re-encryption (PRE) allows a proxy with a re-encryption key \(\textsf{rk}_{A\rightarrow B}\) to transform Alice’s ciphertext to Bob’s ciphertext without revealing the underlying message. Since its introduction, numerous PRE schemes and variants have been studied, but almost all of them assume that both Alice and Bob use a public-key encryption (PKE) system. However, it is more likely that Alice and Bob use different encryption systems like identity-based encryption (IBE), attribute-based encryption (ABE) and functional encryption (FE). This limitation restricts the broader applicability of PRE. In particular, Döttling and Nishimaki [PKC 2021] leave defining and realizing re-encryptions between different primitives as an open problem. In this paper, we explore the feasibility of re-encryptions across different encryption systems. To this end, we first define a primitive named generalized functional encryption (GFE) that unifies PKE, IBE, ABE and FE, and formalize the syntax and security models for re-encryptions from one GFE to another GFE ( \(\textsf{GFE}_1\rightarrow \textsf{GFE}_2\) ). Then, we present two generic constructions of \(\textsf{GFE}_1\rightarrow \textsf{GFE}_2\) and show how to instantiate them. As a complement, we also propose a generic construction of PKE/IBE/ ABE/FE \(\rightarrow \) PKE/IBE/ABE from garbled circuits, by extending the techniques in [Döttling and Nishimaki, PKC 2021]. This supports arbitrary PKE, IBE, ABE, FE schemes, but only achieves non-fine-grained re-encryption. Overall, our (fine-gained) re-encryptions \(\textsf{GFE}_1\rightarrow \textsf{GFE}_2\) allow Alice and Bob to use different encryption systems, broadening the applicability of re-encryption techniques to real-world scenarios, and resolving the open problem raised by Döttling and Nishimaki [PKC 2021].