Generative recommendation aims to learn the underlying generative process over the entire item set to produce recommendations for users. Although it leverages non-linear probabilistic models to surpass the limited modeling capacity of linear factor models, it is often constrained by a trade-off between representation ability and tractability. With the rise of a new generation of generative methods based on pre-trained language models (LMs), incorporating LMs into general recommendation with implicit feedback has gained considerable attention. However, adapting them to generative recommendation remains challenging. The core reason lies in the mismatch between the input-output formats and semantics of generative models and LMs, making it challenging to achieve optimal alignment in the feature space. This work addresses this issue by proposing a model-agnostic generative recommendation framework called DMRec, which introduces a probabilistic meta-network to bridge the outputs of LMs with user interactions, thereby enabling an equivalent probabilistic modeling process. Subsequently, we design three cross-space distribution matching processes aimed at maximizing shared information while preserving the unique semantics of each space and filtering out irrelevant information. We apply DMRec to three different types of generative recommendation methods and conduct extensive experiments on three public datasets. The experimental results demonstrate that DMRec can effectively enhance the recommendation performance of these generative models, and it shows significant advantages over mainstream LM-enhanced recommendation methods.