Images of subset of finite set under iterations of random mappings

Let $\mathcal{N}$ be a set of $N$ elements and $F_1,F_2,\ldots$ be a sequence of random independent equiprobable mappings $\mathcal{N}\to\mathcal{N}$. For a subset $S_0\subset \mathcal{N},\,|S_0|=n$, we consider a sequence of its images $S_k=F_k(\ldots F_2(F_1(S_0))\ldots),\,k=1,2\ldots$, and a sequence of their unions $\Psi_k=S_1\cup\ldots\cup S_k,\,k=1,2\ldots$  An approach to the exact computation of distribution of $|S_k|$ and $|\Psi_k|$ for moderate values of $N$ is described. Two-sided inequalities for $\mathbf{M}|S_k|$ and $\mathbf{M}|\Psi_k|$ such that upper bound are asymptotically equivalent to lower ones for $N,n,k\to\infty, nk=o(N)$ are derived. The results are of interest for the analysis of time-memory tradeoff algorithms.