Given the immense progress in the study of classical and synthetic Ricci lower bounds in recent decades, it is desirable to search for analogous results in the parabolic, time-evolving setting of Ricci flows. Many nice analytic/geometric properties known to hold for Ricci flows, such as Perelman's W and reduced volume monotonicities, can be recovered assuming only an inhomogeneous quadratic tensor inequality \(D \geq 0\), introduced by Buzano in 2010. For this and other heuristic reasons,  \(D \geq 0\) can be interpreted as a nonnegative curvature condition in this setting. This talk will present recent work proving the equivalence of \(D \geq 0\) to a large number of monotonicity/convexity statements related to the heat flow and 
optimal transport on a smoothly evolving family of (weighted) Riemannian manifolds. Some of these statements were previously known to hold under \(D \geq 0\) while others are new even for Ricci flows, but each of them has a strict analog characterizing \(\mathrm{Ric} \geq 0\) (and dimension \(\leq N \leq \infty\)) in the constant-in-time setting. Everything is joint work with Marco Flaim.