Excitable reaction-diffusion systems form a wealth of dissipative concentration patterns that exist not only in chemical systems but also control or disrupt biological functions. An important example are rotating spiral waves in the autocatalytic Belousov-Zhabotinsky reaction. We show that the viscosity of this system can be increased by the addition of the polymer xanthan gum. In the resulting system, we pin spiral waves to a thin glass rod and then reposition the vortex centers by a linear motion of the heterogeneity. The Stokes flow generated by this motion can be a weak perturbation to the wave pattern and follows a simple, analytical expression. Numerical simulations of a corresponding reaction-diffusion-flow model reproduce the experimental observations and show that the spatial extent of the flow field can vary widely around the characteristic wavelength of the spiral. We find that a sharp spatial decay of the flow pattern corresponds to our experimental observations, whereas more expansive flow fields surprisingly allow the repositioning of spiral tips at speeds faster than the wave velocity.