Optical and photoemission measurements were performed on alkali antimonide ${\mathrm{Na}}_{2}\mathrm{KSb}$ and ${\mathrm{Na}}_{2}\mathrm{KSb}/{\mathrm{Cs}}_{x}\mathrm{Sb}$ photocathodes in order to determine their energy-band diagrams, elucidate the photoemission pathways, and explore the options for interface engineering in order to reach high quantum efficiencies of the photocathodes. This study is motivated by the recent discovery of optical orientation in ${\mathrm{Na}}_{2}\mathrm{KSb}$ and emission of spin-polarized electrons from ${\mathrm{Na}}_{2}\mathrm{KSb}/{\mathrm{Cs}}_{x}\mathrm{Sb}$ photocathodes [V.S. Rusetsky et al., Phys. Rev. Lett. 129, 166802 (2022)]. We have shown that the band gap ${E}_{g}$ of ${\mathrm{Na}}_{2}\mathrm{KSb}$ at $T=295$ K lies within the range of 1.40--1.44 eV. The ${\mathrm{Na}}_{2}\mathrm{KSb}$ surface activation by the deposition of $\mathrm{Cs}$ and $\mathrm{Sb}$ results in effective electron affinity decrease by approximately 0.37 eV, and in an increase of the quantum efficiency up to 0.2 electrons per incident photon. The analysis of longitudinal energy distribution curves (EDCs) proves that the surface of activated ${\mathrm{Na}}_{2}\mathrm{KSb}/{\mathrm{Cs}}_{x}\mathrm{Sb}$ photocathodes have negative effective electron affinity of approximately $\ensuremath{-}0.1$ and $\ensuremath{-}0.25$ eV at $T=295$ and 80 K, respectively. EDC measurements under increasing photon energy $\ensuremath{\hbar}\ensuremath{\omega}$ demonstrate the transition of photoemission pathway from the surface states' photoionization at $\ensuremath{\hbar}\ensuremath{\omega}<{E}_{g}$ to the emission from the conduction-band bottom at $\ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\approx}{E}_{g}$ and from the states with high kinetic energy in the conduction band at $\ensuremath{\hbar}\ensuremath{\omega}>{E}_{g}$. EDCs measured at 80 K reveal a highly directional photoelectron emission from the ${\mathrm{Na}}_{2}\mathrm{KSb}/{\mathrm{Cs}}_{x}\mathrm{Sb}$ photocathode, as compared to the p-$\mathrm{Ga}\mathrm{As}$($\mathrm{Cs}$,$\mathrm{O}$) photocathode. This fact, along with the observed significant, by an order of magnitude, increase in the photoluminescence intensity under the ${\mathrm{Na}}_{2}\mathrm{KSb}$ surface activation by $\mathrm{Cs}$ and $\mathrm{Sb}$, indicates relatively weak diffuse scattering in the ``quasiepitaxial'' ${\mathrm{Cs}}_{x}\mathrm{Sb}$ activation layer of a ${\mathrm{Na}}_{2}\mathrm{KSb}/{\mathrm{Cs}}_{x}\mathrm{Sb}$ photocathode, compared to strong scattering in the amorphous ($\mathrm{Cs}$,$\mathrm{O}$) activation layer of a p-$\mathrm{Ga}\mathrm{As}$($\mathrm{Cs}$,$\mathrm{O}$) photocathode.