Characterizing the nature of many-body localization transitions (MBLTs) and their potential critical behaviors has remained a challenging problem. In this work, we study the dynamics of the displacement, quantifying the spread of the radial probability distribution in the Fock space, for systems with MBLTs, and perform a finite-size scaling analysis. We find that the scaling exponents satisfy theoretical bounds, and can identify universality classes. We show that reliable extrapolations to the thermodynamic limit for the MBLT induced by quasiperiodic fields is possible even for computationally accessible system sizes. Our work highlights that the displacement is a valuable tool for studying MBLTs, as relevant to ongoing experimental efforts.