The outstanding photovoltaic performances of lead halide perovskite (LHP) thin film solar cells are due in particular to the large diffusion length of photocarriers. The mechanism behind this property and its dependence on the various anions and cations in the LHP material composition are still an active topic of debate. Here, we apply ultrabroadband terahertz spectroscopy with a time resolution < 200 fs to probe the early carrier mobility dynamics of photoexcited perovskite samples of different chemical compositions. An increase in the carrier mobility with time constants in the range of 0.3 to 0.7 ps is observed for different LHPs, in which the signal amplitude is larger at shorter excitation wavelengths. This feature is assigned to the relaxation of hot carriers from low-curvature regions of the band structure to the bottom of the valley, where their effective mass accounts for the maximum mobility. The dependence of the mobility dynamics on the initial photogenerated carrier population shows that carrier cooling competes with a dynamic screening effect associated with polaron formation and relaxation. A kinetic model that considers all competing processes is proposed. Simulations based on this model allow us to fit the experimental data well and evidence a composition dependence of the dynamic screening lifetime.