Ultracold atomic gases in optical lattices are an ideal platform for studying quantum many-body physics. The long timescales and isolated nature of these systems makes them particularly suited for exploring the dynamics of nearly closed quantum systems and their relaxation towards thermal equilibrium. In this thesis, we demonstrate the realization of two novel cold atom systems: lattice Fermi gases with non-local interactions and tilted Fermi-Hubbard systems. In both of these systems, we explore the slow relaxation of density perturbations, either due to kinetic constraints or unusual hydrodynamics. The first system we study is a Fermi gas laser coupled to a Rydberg state. For near resonant coupling of a localized gas in a unit-filled lattice, we realize a quantum Ising model with transverse and longitudinal fields. We study the out-of-equilibrium dynamics of antiferromagnetic correlations in this spin system. For far off-resonant Rydberg coupling, we prepare itinerant Fermi gases with strong non-local interactions. In this Rydberg-dressed regime, we introduce a small Rydberg admixture to the ground state of the system which results in a laser-tunable soft-core interaction potential. We use this technique to realize a $t-V$ model with spin-polarized fermions and study the dynamics of imprinted charge density waves. For strong off-site interactions, the number of bonds is approximately conserved, which leads to slow relaxation of these states. More generally, the Rydberg-dressing technique is promising for future studies of extended Hubbard models in multi-component systems. The second system we study is the two-dimensional Fermi-Hubbard model in the presence of a large tilt. When the tilt is aligned with a lattice axis, the system exhibits slow thermalization and subdiffusive charge transport due to modified hydrodynamics where heat transport acts as a bottleneck for charge transport. This work sets the stage for studying the complete breakdown of thermalization expected for more generic tilt angles where Hilbert space fragmentation is expected.