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Abstract We find the scaling limits of a general class of boundary-to-boundary connection probabilities and multiple interfaces in the critical planar FK-Ising model, thus verifying predictions from the physics literature. We also discuss conjectural formulas using Coulomb gas integrals for the corresponding quantities in general critical planar random-cluster models with cluster-weight$${q \in [1,4)}$$ $q\in [1,4)$. Thus far, proofs for convergence, including ours, rely on discrete complex analysis techniques and are beyond reach for other values ofqthan the FK-Ising model ($$q=2$$ $q=2$). Given the convergence of interfaces, the conjectural formulas for other values ofqcould be verified similarly with relatively minor technical work. The limit interfaces are variants of$$\text {SLE}_\kappa $$ ${\text{SLE}}_{\kappa }$curves (with$$\kappa = 16/3$$ $\kappa =16/3$for$$q=2$$ $q=2$). Their partition functions, that give the connection probabilities, also satisfy properties predicted for correlation functions in conformal field theory (CFT), expected to describe scaling limits of critical random-cluster models. We verify these properties for all$$q \in [1,4)$$ $q\in [1,4)$, thus providing further evidence of the expected CFT description of these models.