Integral preserving schemes for ODEs can be derived by means of for instance discrete gradient methods. For PDEs, one may first discretize in space, using for instance finite difference methods or finite element methods, and then apply an integral preserving method for the corresponding ODEs. For PDEs discretized on moving grids, the situation is more complicated, it is not even clear exactly what should be meant by an integral preserving scheme in this setting. We shall propose a definition and then derive the resulting conservative schemes, both with finite difference schemes and with finite element schemes. We test the methods on problems with travelling wave solutions and demonstrate that they give remarkably good results, both compared to fixed grid and to non-conservative schemes. This is joint work with S. Eidnes and T. Ringholm.