We consider the unitary dynamics of interacting fermions in the lowest Landau level, on spherical and toroidal geometries. The dynamics are driven by the interaction Hamiltonian which, viewed in the basis of single-particle Landau orbitals, contains correlated pair hopping terms in addition to static repulsion. This setting and this type of Hamiltonian has a significant history in numerical studies of fractional quantum Hall (FQH) physics, but the many-body quantum dynamics generated by such correlated hopping has not been explored in detail. We focus on initial states containing all the fermions in one block of orbitals. We characterize in detail how the fermionic liquid spreads out starting from such a state. We identify and explain differences with regular (single-particle) hopping Hamiltonians. Such differences are seen, e.g. in the entanglement dynamics, in that some initial block states are frozen or near-frozen, and in density gradients persisting in long-time equilibrated states. Examining the level spacing statistics, we show that the most common Hamiltonians used in FQH physics are not integrable, and explain that GOE statistics (level statistics corresponding to the Gaussian orthogonal ensemble) can appear in many cases despite the lack of time-reversal symmetry.