We discuss how to perform interpolations between relativistic and static computations in order to extract heavy-light B-physics observables in the continuum. This strategy can be carried out entirely in large volume, but its predictivity is enhanced by the following step scaling approach. Relativistic computations are carried out at the physical b-quark mass using the Schrödinger Functional in a box, where small $am$ is accessible. They are connected to large volume observables through step scaling functions that trace the mass dependence between the physical charm region and the static limit, such that B-physics results can be obtained by interpolation. We discuss the extraction of relativistic large volume observables entering the computation, focusing on leptonic and semi-leptonic transition amplitudes. We review in detail our numerical results first presented in [1] for leptonic decays from CLS $N_f=2+1$ ensembles at $m_u=m_d=m_s$, and with five values of the lattice spacing down to $0.039 \ \mathrm{fm}$.