  This paper studies the critical and near-critical regimes of the planar
random-cluster model on $\mathbb Z^2$ with cluster-weight $q\in[1,4]$ using
novel coupling techniques. More precisely, we derive the scaling relations
between the critical exponents $\beta$, $\gamma$, $\delta$, $\eta$, $\nu$,
$\zeta$ as well as $\alpha$ (when $\alpha\ge0$). As a key input, we show the
stability of crossing probabilities in the near-critical regime using new
interpretations of the notion of influence of an edge in terms of the rate of
mixing. As a byproduct, we derive a generalization of Kesten's classical
scaling relation for Bernoulli percolation involving the ``mixing rate''
critical exponent $\iota$ replacing the four-arm event exponent $\xi_4$.
