  The main goal in this paper is to study asymptotic behaviour in
$L^p(\mathbb{R}^N)$ for the solutions of the fractional version of the discrete
in time $N$-dimensional diffusion equation, which involves the Caputo
fractional $h$-difference operator. The techniques to prove the results are
based in new subordination formulas involving the discrete in time Gaussian
kernel, and which are defined via an analogue in discrete time setting of the
scaled Wright functions. Moreover, we get an equivalent representation of that
subordination formula by Fox H-functions.
