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The goal of this paper is twofold: we study metric measure spaces $(X,d,m)$with variable lower bounds for the Ricci curvature and we study pathwisecoupling of Brownian motions. Given any lower semicontinuous function $k:X\to\mathbb R$ we introduce the curvature-dimension condition $CD(k,\infty)$ whichcanonically extends the curvature-dimension condition $CD(K,\infty)$ ofLott-Sturm-Villani for constant $K\in \mathbb R$. For infinitesimallyHilbertian spaces we prove i) its equivalence to an evolution-variationinequality $EVI_k$ which in turn extends the $EVI_K$-inequality ofAmbrosio-Gigli-Savar\'e; ii) its stability under convergence and itslocal-to-global property.  For metric measure spaces with uniform lower curvature bounds $K$ we provethat for each pair of initial distributions $\mu_1,\mu_2$ on $X$ there exists acoupling $B_t=(B_t^1,B_t^2)$, $t\ge0$, of two Brownian motions on $X$ with thegiven initial distributions such that a.s. for all $s,t\ge0$  $$d(B^1_{s+t},B^2_{s+t})\le e^{-K t/2}\cdot d(B_s^1,B_s^2).$$

Metric Measure Spaces with Variable Ricci Bounds and Couplings of Brownian Motions