A grid approximation of a boundary value problem is considered for a singularly perturbed parabolic reaction-diffusion equation in domains with moving boundaries. Using widths similar to Kolmogorov’s widths, necessary and sufficient conditions for the ε-uniform convergence of approximations to the solution of the boundary value problem are established. Taking into account these conditions, a difference scheme is constructed on a mesh which is piecewise uniform in a coordinate system adapted to the moving boundary. This scheme converges ε-uniformly in the maximum norm. Ability to apply the technique based on the widths for an investigation of ε-uniformly convergent difference schemes is discussed.