We consider discrete-time fast-slow systems of the form $$ X^{(n)}_{k+1} =
X^{(n)}_k + n^{-1}a_n(X_k^{(n)},Y_k^{(n)}) +
n^{-1/2}b_n(X_k^{(n)},Y_k^{(n)})\;, \quad Y_{k+1}^{(n)} = T_nY_k^{(n)}\;.$$ We
give conditions under which the dynamics of the slow equations converge weakly
to an It\^o diffusion $X$ as $n\to\infty$. The drift and diffusion coefficients
of the limiting stochastic differential equation satisfied by $X$ are given
explicitly. This extends the results of [Kelly--Melbourne, J. Funct. Anal. 272
(2017) 4063-4102] from the continuous-time case to the discrete-time case.
Moreover, our methods ($p$-variation rough paths) work under optimal moment
assumptions.