  In this paper we study random flights in R^d with displacements possessing
Dirichlet distributions of two different types and uniformly oriented. The
randomization of the number of displacements has the form of a generalized
Poisson process whose parameters depend on the dimension d. We prove that the
distributions of the point X(t) and Y(t), t \geq 0, performing the random
flights (with the first and second form of Dirichlet intertimes) are related to
Klein-Gordon-type partial differential equations. Our analysis is based on
McBride theory of integer powers of hyper-Bessel operators. A special attention
is devoted to the three-dimensional case where we are able to obtain the
explicit form of the equations governing the law of X(t) and Y(t). In
particular we show that that the distribution of Y(t) satisfies a
telegraph-type equation with time-varying coefficients.
