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 Teichmüller 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 Teichmüller space of $M$, and we relate the growth of the speed to the Teichmüller distance to a basepoint. One key argument is an adaptation of Gouëzel's pivoting techniques to actions of a fixed group on a sequence of hyperbolic metric spaces.