Move just some points of the finite-element mesh (triangle/tetrahedron) and interpolate the solution in the new mesh

I have a finite element mesh, and I want that the mesh has some specific points and edges, as I show in the picture. I think that that is possible in a mesh software.

I’m solving a evolutive (time-dependent) PDE using Finite Element Method. The blue line going to move, so I want to generate in each time step a mesh containing these points and edges in the new position. The rest of the mesh (points and edges) could be the same.

The problem is that calculate the solution in each vertex of the new mesh is expensive, but because I need to move just some few points I think that maybe exists some algorithm that allows me to move just the blue points. Of course, the new meshes must be good meshes for FEM, so it is not just to move the points, some of the elements must disappear and other must be created (to have elements with minimal angles…), but I think that the majority of the mesh could be the same, and that is why I think that could be a smarter algorithm to create the new meshes and calculate the solutions in the new mesh. My preference is to use triangles and tetrahedra, in 2D and 3D resp.

Do you have some experience or comment that you think that could me help?