Interactive rendering of soft shadows (or penumbra) in scenes with moving objects is a challenging problem. High quality walkthrough rendering of static scenes with penumbra can be achieved using pre-calculated discontinuity meshes, which provide a triangulation well adapted to penumbral boundaries, and backprojections which provide exact illumination computation at vertices very efficiently. However, recomputation of the complete mesh and backprojection structures at each frame is prohibitively expensive in environments with changing geometry. This recomputation would in any case be wasteful: only a limited part of these structures actually needs to be recalculated. We present a novel algorithm which uses spatial coherence of movement as well as the rich visibility information existing in the discontinuity mesh to avoid unnecessary recomputation after object motion. In particular we isolate all modifications required for the update of the discontinuity mesh by using an augmented spatial subdivision structure and we restrict intersections of discontinuity surfaces with the scene. In addition, we develop an algorithm which identifies visibility changes by exploiting information contained in the planar discontinuity mesh of each scene polygon, obviating the need for many expensive searches in 3D space. A full implementation of the algorithm is presented, which allows interactive updates of high-quality soft shadows for scenes of moderate complexity. The algorithm can also be directly applied to global illumination.