In the first part of the paper, we consider a “random flight” process in R^d and obtain the weak limits under different transformations of the Poissonian switching times. In the second part, we construct diffusion approximations for this process and investigate their accuracy. To prove the weak convergence result, we use the approach of [15]. We consider more general model which may be called “random walk over ellipsoids in R^d ”. For this model, we establish the Edgeworth-type expansion. The main tool in this part is the parametrix method [5, 7].