Recent experimental data on several observables in semileptonic $B$-meson decays are found to be in tension with the corresponding Standard Model predictions. Most of these deviations are related to $b \to c$ and $b \to s$ flavour changing transitions. In this work, we estimate possible New Physics effects in $b \to d \mu^+ \mu^-$ flavour changing neutral currents. We parametrize NP contributions in a model-independent way and determine the allowed ranges of corresponding Wilson coefficients from the data on the exclusive $B^\pm \to \pi^\pm \mu^+ \mu^-$ decays measured recently by the LHCb collaboration. Afterwards, we investigate the impact of these results on other $b \to d$ processes such as the leptonic $B^0 \to \mu^+ \mu^-$ decays and $B^0 - \bar B^0$ mixing. As an example, we consider a simplified $Z^\prime$ model that is found to be consistent with current $b \to d$ data in the certain regions of the NP parameter space. In addition, we estimate the correlations between the partial decay widths of $B \to \pi \mu^+ \mu^-$ and $B \to K \mu^+ \mu^-$ processes to be used for an independent determination of CKM matrix elements as well as for a combined New Physics analysis of both $b \to d$ and $b \to s$ transitions.