We present the status of an ongoing lattice-QCD calculation of form factors for exclusive semileptonic decays of $B$ mesons with both charged currents ($B\to\pi\ell\nu$, $B_s\to K\ell\nu$) and neutral currents ($B\to\pi\ell^+\ell^-$, $B\to K\ell^+\ell^-$). The results are important for constraining or revealing physics beyond the Standard Model. This work uses MILC's (2+1+1)-flavor ensembles, with the HISQ action for the sea and light valence quarks, and the clover action in the Fermilab interpretation for the $b$ quark. Simulations are carried out at four lattice spacings down to 0.088 fm, with both physical and a few nonphysical sea-quark masses. We present preliminary results for correlation-function fits and the chiral-continuum extrapolation of the form factors.