Model theoretic forcing is extended to wider languages with some applications, for example:

1- Model theoretic forcing in analysis which gives an exposition of forcing for metric structures.

2- Shelah extended it to the context of abstract elementary classes, see for example his book Classiﬁcation theory for abstract elementary classes, Section IV.2.

You may also look at Vasey's thesis Superstability and Categoricity in Abstract Elementary Classes.

3- Andres Villaveces has several papers, where he defines some variants of forcing. See for example Sheaves of G-structures and generic G-models and Sheavesofmetricstructures and ...

4- You may be also interested in the work of Xavier Caicedo Logic of sheaves of structures. In this paper, Caicedo introduces a concept of sheaves of structures over a topological space, and a notion of model-theoretic forcing for such structures. This is a very general approach to model-theoretic forcing. For an English exposition, you may look at The logic of sheaves, sheaf forcing and the independence of the Continuum Hypothesis.