Full Heusler alloys of the series ${\mathrm{Fe}}_{2\text{\ensuremath{-}}\mathrm{x}}{\mathrm{Ni}}_{\mathrm{x}}\mathrm{VAl},0\ensuremath{\le}x\ensuremath{\le}0.2$, were prepared and characterized, and their physical properties, relevant to the thermoelectric performance of such materials, were studied in a wide temperature range. The starting material ${\mathrm{Fe}}_{2}\mathrm{VAl}$ is characterized by a pseudogap of the electronic density of states near the Fermi energy, with a gap width of the order of 1 eV. Density functional theory calculations were performed by application of two approaches. In the framework of the local-spin-density approximation and coherent potential approximation, the electronic densities of states of substitutional alloys were calculated, revealing that with increasing Ni content the Fermi energy moves toward the conduction band, and consequently, the nature of electronic transport changes from $p$ type to $n$ type. It appears that Ni, due to its extra electrons, provides a narrow impurity band near the Fermi level. These states can be made responsible for the experimentally observed evolution of transport properties. Furthermore, the Vienna ab initio Simulation package (vasp) was utilized for deriving electronic, structural, and vibrational properties of ordered ${\mathrm{Fe}}_{2}\mathrm{VAl}$ and ${\mathrm{Fe}}_{1.75}{\mathrm{Ni}}_{0.25}\mathrm{VAl}$. In particular, it is found that due to Ni substitution there is a general shift to lower phonon frequencies by about 2 THz as compared to the undoped case. Associated to these modifications, the electrical resistivity, $\ensuremath{\rho}(T)$, changes from a semiconducting-like behavior to a nonsimple metallic behavior, while the Seebeck coefficient reaches values of the order of $\ensuremath{-}80$ $\ensuremath{\mu}\mathrm{V}$/K around room temperature for the sample $x=0.2$. The increase of the Ni content, in addition, goes along with a substantial reduction of the lattice part of the thermal conductivity. This change is analyzed in detail in terms of a disorder parameter $\mathrm{\ensuremath{\Gamma}}$, characterizing the derangement of the crystalline lattice due to the substitution of Fe by Ni. Ab initio calculations of the phonon dynamics carried out for ${\mathrm{Fe}}_{2}\mathrm{VAl}$ and for ${\mathrm{Fe}}_{1.75}{\mathrm{Ni}}_{0.25}\mathrm{VAl}$ support these analyses.