Laser sources have become indispensable for industrial materials processing applications. These applications are accompanied with a variety of different demands and requirements on the delivered laser irradiance distributions. With a high spatial uniformity, top-hat beams provide benefits for applications like surface heat treatment or welding, in which it is desirable to uniformly illuminate a target surface. Some applications might not only favor a specific beam irradiance distribution but can benefit additionally from time-varying distributions. In this work, we present the analytic design of an XY-zoom beam expander based on movable freeform optics that allows to simultaneously vary the magnification in x- and y-direction, respectively. This optical functionality is not new; what is new is the idea that axially moving freeform lenses are used to achieve such an optical functionality by optimally exploiting the additional degrees of freedom that freeform surfaces offer. The developed analytic solution is fully described by very few initial parameters and does allow an increasingly accurate calculation of four freeform lenses described by high order XY Taylor polynomial surfaces. Moreover, this solution approach can be adapted to cope with additional optical surfaces and/or lens groups to further enhance the overall optical performance. In comparison with (existing) combinations of rotated cylindrically symmetric zoom beam expanders, such a freeform system consists of less optical elements and provides a much more compact solution, yet achieving excellent overall optical performance throughout the full range of zoom positions.