Semi-inclusive neutrino-nucleus cross sections within the plane-wave impulse approximation (PWIA) for three nuclear models: relativistic Fermi gas (RFG), independent-particle shell model (IPSM) and natural orbitals shell model (NO) are compared with the available CC0$\pi$ measurements from the T2K, MINER$\nu$A and MicroBooNE collaborations where a muon and at least one proton were detected in the final state. Results are presented as a function of the momenta and angles of the final particles, as well as in terms of the imbalances between proton and muon kinematics. The analysis reveals that contributions beyond PWIA are crucial to explain the experimental measurements and that the study of correlations between final-state proton and muon kinematics can provide valuable information on relevant nuclear effects such as initial state dynamics and final state interactions.