This question is difficult to answer in general terms because the framework chosen is highly dependent on the orientation of single scholars. There are still some hard-core Chomskians left and the principles of GG are still taught at Uni in order for students to understand and describe phenomena of constituency and dependency. However, in the last few years, there has been a consistent shift towards adopting the tenets of Construction Grammar a la Goldberg (1995) https://books.google.de/books/about/Constructions.html?id=HzmGM0qCKtIC&redir_esc=y
This approach focuses on existent constructions as forms with a certain syntactic configuration and pairs it with a semantic side, their meaning. There are two main pros related to it: (1) It reflects the way speakers produce utterances on the basis of constructions that are stored in their mental inventory by way of being used in a certain language. We use language through routinized blocks that we have heard lots of times, so to say. We do not proceed by shifting constituents or cumulate them in order to produce a sentence (contrary to Chomsky). This would be unrealistic in terms of language production. (2) It accounts for the fact that syntax is tightly linked with semantics. The latter played a minimal role in Chomsy's framework, if at all.
