  The form factors of semileptonic $B_{(s)}\to a_{1}(K_1)\, \ell^{+}\ell^{-}$,
$ \ell=\tau,\mu,e$ transitions are investigated in the framework of the
light-cone sum rules with $B$-meson distribution amplitudes, which play an
important role in exclusive $B$ decays. The $B$-meson distribution amplitudes,
$\varphi_{\pm}(\omega)$ are a model-dependent form, so we consider four
different parameterizations which can provide a reasonable description of
$\varphi_{\pm}(\omega)$ from QCD corrections. The branching fractions of these
transitions are calculated. For a better analysis, a comparison of our results
with the prediction of other models is provided.
