We consider several classes of $sigma$-models (on groups and symmetric spaces, $eta$-models, $lambda$-models) with local couplings that may depend on the 2d coordinates, e.g. on time $tau$. We observe that (i) starting with a classically integrable 2d $sigma$-model, (ii) formally promoting its couplings $h_alpha$ to functions $h_alpha(tau)$ of 2d time, and (iii) demanding that the resulting time-dependent model also admits a Lax connection implies that $h_alpha(tau)$ must solve the 1-loop RG equations of the original theory with $tau$ interpreted as RG time. This provides a novel example of an integrability - RG flow connection. The existence of a Lax connection suggests that these time-dependent $sigma$-models may themselves be understood as integrable. We investigate this question by studying the possibility of constructing non-local and local conserved charges. Such $sigma$-models with $D$-dimensional target space and time-dependent couplings subject to the RG flow naturally appear in string theory upon fixing the light-cone gauge in a $(D+2)$-dimensional conformal $sigma$-model with a metric admitting a covariantly constant null Killing vector and a dilaton linear in the null coordinate.
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