We present a series of applications of the Jacobi evolution equations along geodesics in groups of diffeomorphisms. We describe, in particular, how they can be used to perform feasible gradient descent algorithms for image matching, in several situations, and illustrate this with 2D and 3D experiments. We also discuss parallel translation in the group, with its projections on shape manifolds, and focus in particular on an implementation of the associated equations using iterated Jacobi fields.