  We report on a calculation of form factors for the semileptonic decay of $B$
meson to pion on $2+1$-flavour lattices with lattice spacings from 0.080 fm
down to 0.044 fm. Using the M\"obius domain wall fermion action for both sea
and valence quarks, we simulate pions with masses down to 225 MeV. By utilizing
a range of heavy quark masses up to 2.44 times the mass of the charm quark we
extrapolate to the physical $b$ quark mass. We discuss the dependence of the
form factors on the pion mass, heavy quark mass, lattice spacing and the
momentum-transfer. We extract the CKM matrix element $|V_{ub}|$ through a
simultaneous fit with the $B \to \pi\ell\nu$ differential branching fractions
provided by the Belle and BaBar collaborations after a chiral-continuum and
physical $b$ quark extrapolations of our lattice data.
