The capability to reliably program partial polarization states with nanosecond programming speed and femtojoule energies per bit in ferroelectrics makes them an ideal candidate to realize multibit memory elements for high-density crossbar arrays, which could enable neural network models with a large number of parameters at the edge. However, a thorough understanding of the domain switching dynamics involved in the polarization reversal is required to achieve full control of the multibit capability. Transient current integration measurements are adopted to investigate the domain dynamics in aluminum scandium nitride (Al_0.85Sc_0.15N) and hafnium zirconium oxide (Hf_0.5Zr_0.5O₂). The switching dynamics are correlated to the crystal structure of the films. The contributions of domain nucleation and domain wall motion are decoupled by analyzing the rate of change of the time-dependent normalized switched polarization. Thermally activated creep domain wall motion characterizes the Al_0.85Sc_0.15N switching dynamics. The statistics of independently nucleating domains and the domain wall creep motion in Hf_0.5Zr_0.5O₂ are associated with the spatially inhomogeneous distribution of local switching field due to polymorphism, absence of preferential crystallite orientation, as well as defects and charges that can be located at the grain boundaries. The c-axis texture, single-phase nature, and strong likelihood of less fabrication process-induced defects contribute to the homogeneity of the local switching field in Al_0.85Sc_0.15N. Nonetheless, defects generated and redistributed upon bipolar electric field switching cycling result in Al_0.85Sc_0.15N domain wall pinning. The wake-up effect in Hf_0.5Zr_0.5O₂ is explained thorough the continuous addition of switchable regions associated with two independent distributions of characteristic switching times.