When a physical object, which is perceived as a singularity on a certain level of mathematical description, is set into motion, a paradox may arise rendering dynamic description impossible unless the singularity is resolved by introducing new physics in the singular core. This situation, appearing in diverse physical contexts, necessitates application of multiscale matching methods, employing a simpler long-scale model in the far field and a short-scale model with more detailed physical contents in the core of the singularity. The law of motion can be derived within this approach by applying a modified Fredholm alternative in a region large compared to the inner and small compared with the outer scale, and evaluating the boundary terms which determine both the driving force and dissipation. I give examples of applying this technique to both topological (vortices) and non-topological (contact lines) singularities.