Many mathematical models of interest are typically described at three main scales: the microscopic (agent-based), the mesoscopic (kinetic), and the macroscopic (population-based) level. In this talk, we focus on the time-asymptotic behavior and the kinetic-to-macroscopic transition for kinetic equations of the form: partial_t f + (1/epsilon) v · grad_x f = (1/epsilon^2) Q[f], where Q is a linear operator acting on the velocity variable. Our analysis concentrates on three principal cases: the BGK operator, the kinetic Fokker–Planck operator, and the nonlocal kinetic Fokker–Planck operator, along with their fractional variants. The transition from the kinetic scale to the macroscopic diffusive regime is nontrivial, particularly when seeking quantitative rates of convergence, which are often subtle. Previous works have addressed the diffusive limit in various norms for certain classes of linear operators using different techniques, and have also studied time decay in similar settings. Our main contribution is a unified and quantitative approach to both the time-asymptotic behavior and the diffusive limit (as epsilon tends to zero) of these equations, based on a Wild sum representation. This formulation, widely used in the context of the space-homogeneous Boltzmann equation, is rarely applied to purely kinetic equations. It enables us to simultaneously capture both time-asymptotic behavior and the diffusive limit in strong norms (Lp spaces for all p between 1 and infinity). We also discuss the optimality of our decay rates and potential extensions of the method.