A new accelerator complex is being constructed at the Joint Institute for Nuclear Research as a part of the Nuclotron-based Ion Collider fAcility (NICA) project. The goal is to conduct experiments with colliding ion beams (at the first stage of the project) and colliding polarized proton/deuteron beams (at the second stage). Transport beam channels and the systems of beam injection and extraction for synchrotrons and collider rings are an important connecting link for the whole accelerator facility. The design of the primary beam-transport channels and injection/extraction systems are presented. Special attention is paid to various aspects of dynamics of beams in their transfer between the NICA accelerators.