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We compute the complete set of SM and tensor $B_{(s)}\toD_{(s)}^*\ell\bar{\nu}$ semileptonic form factors across the full kinematicrange of the decay using second generation MILC $n_f=2+1+1$ HISQ gluon fieldconfigurations and HISQ valence quarks, with the heavy-HISQ method. Latticespacings range from $0.09\mathrm{fm}$ to $0.044\mathrm{fm}$ with pion massesfrom $\approx 300\mathrm{GeV}$ down to the physical value; currents arenormalised nonperturbatively. Using the recent untagged $B\toD^*\ell\bar{\nu}_\ell$ data from Belle and $B_s\to D_s^*\mu\bar{\nu}_\mu$ fromLHCb together with our form factors we determine a model independent value of$V_{cb}=0.03931(54)_\mathrm{exp}(51)_\mathrm{latt}\times 10^{-3}$, in agreementwith previous exclusive determinations and in tension with the most recentinclusive result at the level of $3.6\sigma$. We also observe a $1.5\sigma$tension between the shape of the differential decay rates computed using ourform factors and those measured by Belle. We compute a purely theoreticalStandard Model value for the ratio of semitauonic and semimuonic decay rates,$R(D^*)=0.279(13)$, which we find to be closer to the recent Belle measurementand HFLAV average than theory predictions using fits to experimentaldifferential rate data for $B\to D^*\ell\bar{\nu}_\ell$. Determining $V_{cb}$from our form factors and the experimental total rate for $B\to D^*\ell\nu$also gives a value in agreement with inclusive results. We compute thelongitudinal polarisation fraction for the semitauonic mode,$F_L^{D^*}=0.395(17)$, which is in tension at the level of $2.2\sigma$ with therecent Belle measurement. Our calculation combines $B\to D^*$ and $B_s\toD_s^*$ lattice results, and we provide an update on our previous latticecomputation of the $B_s\to D_s^*$ form factors. We also give the chiralperturbation theory needed to analyse the tensor form factors.

$B \rightarrow D^*$ vector, axial-vector and tensor form factors for the full $q^2$ range from lattice QCD