  We consider the associated production of electroweak gauge bosons and charm
or beauty quark jets at the LHC using the $k_T$-factorization framework. We
apply the transverse momentum dependent (TMD) parton distributions in a proton
obtained from the Parton Branching (PB) method as well as from the
Catani-Ciafaloni-Fiorani-Marchesini (CCFM) evolution equation. For the PB
approach, our prescription merges the standard leading order ${\cal O}(\alpha
\alpha_s)$ $k_T$-factorization calculations with several tree-level
next-to-leading order ${\cal O}(\alpha \alpha_s^2)$ off-shell production
amplitudes. For the CCFM scenario, our consideration is based on the ${\cal
O}(\alpha \alpha_s^2)$ off-shell gluon-gluon fusion subprocess $g^* g^* \to Z/W
+ Q\bar Q$ and some subleading ${\cal O}(\alpha \alpha_s^2)$ subprocesses
involving quark interactions, taken into account in conventional (collinear)
QCD factorization. We find that the $W + c$ and $Z + b$ cross sections,
calculated within the PB and CCFM-based schemes with the proper choice of
leading and next-to-leading subprocesses, are in good agreement with each
other, thus establishing a correspondence between these two scenarios. A
comparison with the latest LHC experimental data is given and the necessity for
the proper off-shell treatment of the production amplitudes in determination of
the parameters of the TMD parton density is demonstrated.
