We report the thermodynamic behavior and short-range magnetic correlations in the Tb2Zr2O7 and Tb2Zr1.5Ti0.5O7 compounds. The parent compound Tb2Zr2O7 has defect fluorite structure, which evolves toward the pyrochlore phase on Ti doping at the Zr site. There is no long-range magnetic order down to 0.4 K, and a magnetic field dependent spin freezing evolves below ~ 1.3 K and ~ 1.0 K for the parent and doped compounds respectively. The ac susceptibility measurements indicate slow spin relaxation processes below ~ 20 K in both compounds. Inelastic neutron scattering reveals broad diffuse scattering, indicative of short-range correlations at low temperature, owing to local structural distortions and persistent spin fluctuations. Our data show that changing the defect-fluorite to the pyrochlore structure through Zr-doping yields only moderate changes of the magnetism dynamics while the overall qualitative response remains unaffected.