A one-parametric family of conformal mappings from the half-plane onto a family of polygons

We consider a one-parametric family of conformal mappings from the upper half-plane onto the family of the polygons obtained by an angle-preserving shift of one or more vertices of some initial polygon. We obtain some differential equation for the family of mappings and some system of ordinary second-order differential equations for the preimages of the vertices of the family of polygons with Cauchy initial-value conditions.