Understanding the anomalies currently observed in semileptonic $B$ decays requires precise theoretical calculations of hadronic matrix elements, parameterised in terms of form factors. In the physical semileptonic region these form factors can be determined most precisely using lattice QCD, with efforts to compute pseudoscalar to vector form factors on the lattice progressing rapidly in recent years. I will discuss the recent HPQCD calculation of the $B\to D^*$ and $B_s\to D^*_s$ form factors using the heavy-HISQ method, as well as showing preliminary results for an update of our currently available $B_c\to J/\psi$ form factors. Finally, I will show preliminary results for the $B_s\to \phi$ form factors.