Families of rational functions uniformizing Riemann surfaces over the sphere

We deduce differential equation for a smooth one-parametric family of rational functions which maps the Riemann sphere onto a given family of ramified coverings of the sphere. As a corollary, we find a system of ODE for the critical points and the poles of these rational functions; the right-hand sides of the equations also depend on the corresponding critical values. The results are used in the problem of uniformization of given compact simply-connected Riemann surfaces over the sphere. The case of genus one (complex tori) is considered as well.