  The recent introduction of the gradient flow has provided a new tool to probe
the dynamics of quantum field theories. The latest developments have shown how
to use the gradient flow for the exploration of symmetries, and the definition
of the corresponding renormalized Noether currents. In this paper we introduce
infinitesimal translations along the gradient flow for gauge theories, and
study the corresponding Ward identities. This approach is readily generalized
to the case of gauge theories defined on a lattice, where the regulator breaks
translation invariance. The Ward identities in this case lead to a
nonperturbative renormalization of the energy-momentum tensor. We discuss an
application of this method to the study of dilatations and scale invariance on
the lattice.
