  Motivated by the recent observation of lepton universality violation in the
flavour changing charged current transitions $b \to c l \bar{\nu}_l$, we intend
to scrutinize the lepton non-universality effects in rare semileptonic $B$
meson decays involving the quark level transitions $b \to u l \bar{\nu}_l$. In
this regard, we envisage the model-independent approach and consider the
generalized effective Lagrangian in the presence of new physics and constrain
the new parameters by using the experimental branching fractions of $B_u^+ \to
l^+ \nu_l$ and $B^- \to \pi^0 \mu^- \bar \nu_\mu$ processes, where $l=e, \mu,
\tau$. We then estimate the branching ratios and forward-backward asymmetries
of $B_{(s)} \to P (V)l \bar{\nu}_l$ processes, where $P(=K, \pi,
\eta^{(\prime)})$ denotes the pseudoscalar meson and $V(=K^*, \rho)$ is the
vector meson. We also find out various lepton non-universality parameters in
these processes in the presence of new physics.
