Card deck reshuffling

Suppose one has a deck of cards. If one reshuffles it using a discrete probability model (based on combinatorics) one will still have the same probability of getting a particular card on each draw.

This seems to be a little bit naïve for real cards, since reshuffling is somewhat non-random. Suppose you know the ordering of the cards in the deck before reshuffling.

Is it possible to tackle non-randomness of reshuffling to get conditional (on original ordering) probabilities of getting a particular card from the deck?

From one perspective, this is a random walk on the symmetric group to get a random permutation $\pi$. $\pi^{-1}(1)$ is a distribution on $\lbrace 1, 2, ..., n \rbrace$, the probability that each card ends up first. If you apply $\pi$ $k$ times then the distribution of the first card is $\pi^{-k}(1)$.
If you start with a random permutation $\nu$ and then apply $\pi^k$, this produces the random permutation $\pi^k \nu$, and the distribution on the first card is $(\pi^k \nu)^{-1}(1) = \nu^{-1} \pi^{-k}(1)$.