  The hadronic form factors of $B_c$ semileptonic decays to the $P-$wave
charmonium 4-plet can be expressed near the zero-recoil point in terms of
universal functions, performing a systematic expansion in QCD in the relative
velocity of the heavy quarks and in $1/m_Q$. Such functions are independent of
the member of the multiplet involved in the transitions. We present the results
of an NLO calculation up to $O(1/m_Q^2)$ classifying the universal functions at
this order. We work out a set of relations among the form factors of the same
mode and of different modes, which should be reproduced by explicit
calculations, reducing the hadronic uncertainty affecting such channels. The
approach is also helpful to investigate the debated nature of
$\chi_{c1}(3872)$, studying the production in $B_c$ semileptonic decays and
comparing it to the modes involving the other $2P$ charmonia.
