  The level dynamics across the many body localization transition is examined
for XXZ-spin model with a random magnetic field. We compare different scenaria
of parameter dependent motion in the system and consider measures such as level
velocities, curvatures as well as their fidelity susceptibilities. Studying the
ergodic phase of the model we find that the level dynamics does not reveal the
commonly believed universal behavior after rescaling the curvatures by the
level velocity variance. At the same time, distributions of level curvatures
and fidelity susceptibilities coincide with properly rescaled distributions for
Gaussian Orthogonal Ensemble of random matrices. Profound differences exists
depending on way the level dynamics is imposed in the many-body localized phase
of the model in which the level dynamics can be understood with the help of
local integrals of motion.
