Polymeric microgels with core-shell morphology provide promising properties for many applications such as controlled uptake and release of guest nanoparticles. In this work we investigated how the structure and dynamics of the core and the shell in the microgel are coupled using both experimental and computer simulation approaches. The studied core-shell model systems which consist of a collapsed core and a swollen shell (CCSS) and a swollen core and collapsed shell (SCCS) show a different behavior in both structure and dynamics. The intermediate scattering profiles obtained from neutron spin echo (NSE) spectroscopy of CCSS microgels show an initial fast decay similar to that of bare swollen microgels followed by a slow decay similar to that of a purely collapsed microgel. This is also reflected in mesoscale hydrodynamic simulations using the multiparticle collision dynamics method. In the case of CCSS microgels, the decay rate of the intermediate scattering functions shows a crossover from collective diffusive dynamics at low wavenumbers to a Zimm-type dynamics at larger wavenumbers. This is similar to the behavior of a purely swollen microgels. In the case of SCCS microgels, the intermediate scattering profiles from experiment and simulations show a slow dynamics at small as well as large wavenumbers. Studying the dynamics of the individual compartments in the simulated structures suggests that the slower dynamics in SCCS microgels can be attributed to the collective motion of collapsed and aggregated shell parts which form in the periphery of the microgel. Additionally, in both CCSS and SCCS microgels, a slowdown of the dynamics is observed in the swollen compartment compared to the bare swollen microgel, which is a result of the interplay between core and shell compartments.