The Gradient Flow is a smoothing technique that has been extensively studied for its renormalization properties. In combination with the short flow-time expansion, it provides a well-defined renormalization scheme. Within this framework, several lattice-specific challenges, such as mixing with lower-dimensional operators, are either avoided or shifted to the perturbative matching stage of the procedure.

I will first introduce the Gradient Flow formalism and outline the key ideas underlying the small flow-time expansion. I will then present our strategy for determining matrix elements of four-quark operators relevant for neutral meson mixing and heavy-meson lifetimes. While meson mixing is well established on the lattice and provides a stringent validation of the method, a first-principles lattice determination of matrix elements governing heavy-meson lifetimes remains an open problem. I will present results for mesons composed of a charm and a strange quark, together with prospects for extending the calculation to B mesons. Particular emphasis will be placed on the applicability constraints of this approach for the observables under consideration, with the aim of highlighting its potential for broader applications in related studies.