An adaptive meshing technique and solution method is proposed in which a two-dimensional body-fitted multi-block mesh is locally adjusted to arbitrarily embedded boundaries that are not necessarily aligned with the mesh. Not only does this scheme allow for rapid and robust mesh generation involving complex embedded boundaries, it also enables the solution of unsteady flow problems involving bodies and interfaces moving relative to the flow domain. This scheme has been implemented within a block-based adaptive mesh refinement (AMR) numerical framework which can ease computational expense while maintaining a detailed representation of the embedded boundary and providing an accurate resolution of the spatial characteristics of the fluid flow. Rigid body motion and evolving motion due to physical processes are considered. A block-based AMR level set method is used to deal with evolving embedded boundaries. Numerical results for various test problems are presented to verify the validity of the scheme as well as demonstrate the capabilities of the approach for predicting complex two-dimensional inviscid and laminar fluid flows.