  Consider a closed surface $M$ with negative Euler characteristic, and an
admissible probability measure on the fundamental group of $M$ with finite
first moment. Corresponding to each point in the Teichm\"uller space of $M$,
there is an associated random walk on the hyperbolic plane. We show that the
speed of this random walk is a proper function on the Teichm\"uller space of
$M$, and we relate the growth of the speed to the Teichm\"uller distance to a
basepoint. One key argument is an adaptation of Gou\"ezel's pivoting techniques
to actions of a fixed group on a sequence of hyperbolic metric spaces.
