  A deck of $n$ cards is shuffled by repeatedly moving the top card to one of
the bottom $k_n$ positions uniformly at random. We give upper and lower bounds
on the total variation mixing time for this shuffle as $k_n$ ranges from a
constant to $n$. We also consider a symmetric variant of this shuffle in which
at each step either the top card is randomly inserted into the bottom $k_n$
positions or a random card from the bottom $k_n$ positions is moved to the top.
For this reversible shuffle we derive bounds on the $L^2$ mixing time. Finally,
we transfer mixing time estimates for the above shuffles to the lazy top to
bottom-$k$ walks that move with probability 1/2 at each step.
