  We present a detailed analysis of random motions moving in higher spaces with
a natural number of velocities. In the case of the so-called minimal random
dynamics, under some wide assumptions, we show the joint distribution of the
position of the motion (both for the inner part and the border of the support)
and the number of displacements performed with each velocity. Explicit results
for cyclic and complete motions are derived. We establish useful relationships
between motions moving in different spaces and we derive the form of the
distribution of the movements in arbitrary dimension. Finally, we investigate
further properties for stochastic motions governed by non-homogeneous Poisson
processes.
