A common motif in biology is the arrangement of cells into tube-like sheets, which further transform into more complex forms. Traditionally, analysis of tissues’ dynamic surfaces has relied on inspecting static snapshots, live imaging of cross-sections, or tracking isolated cells in 3D. However, capturing the interplay between in-plane and out-of-plane behaviors requires following the full surface as it deforms and integrating cell-scale motions into collective, tissue-scale deformations. Here, we introduce an approach to build in toto maps of surface deformation, following tissue parcels in the material frame of reference. The Tube-like sUrface Lagrangian Analysis Resource (‘TubULAR’) provides an open-source MATLAB implementation whose functionality is accessible either as a standalone toolkit or as an extension of the ImSAnE package used in the developmental biology community. TubULAR provides a framework for linking in-plane and out-of-plane behaviors and decomposing complex deformation maps into elementary contributions defining physically meaningful signatures of motion. We underscore the power of our approach by analyzing shape change in the embryonic Drosophila midgut and beating zebrafish heart. In the midgut, our results link asymmetric constrictions to the increasing organ length and to patterned vorticity in the organ chambers. We then extract the dynamics of the beating zebrafish heart, constructing a mode-based description for the organ’s beat cycle. The method naturally generalizes to in vitro and synthetic systems, such as stem cell models of tube development and deforming phase-separated interfaces.