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 $\mathcal{O}(\ensuremath{\alpha}{\ensuremath{\alpha}}_{s})$ ${k}_{T}$-factorization calculations with several tree-level next-to-leading order $\mathcal{O}(\ensuremath{\alpha}{\ensuremath{\alpha}}_{s}^{2})$ off-shell production amplitudes. For the CCFM scenario, our consideration is based on the $\mathcal{O}(\ensuremath{\alpha}{\ensuremath{\alpha}}_{s}^{2})$ off-shell gluon-gluon fusion subprocess ${g}^{*}{g}^{*}\ensuremath{\rightarrow}Z/W+Q\overline{Q}$ and some $\mathcal{O}(\ensuremath{\alpha}{\ensuremath{\alpha}}_{s}^{2})$ 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.