We present the results of an investigation on the pressure behavior of structural, bonding and vibrational properties of $\ensuremath{\beta}$- and $\ensuremath{\gamma}\text{\ensuremath{-}}{\mathrm{C}}_{3}{\mathrm{N}}_{4}$ phases. Emphasis is focused on the trends of the calculated properties along the ${A}_{3}{\mathrm{N}}_{4}$ ($A$: C, Si, Ge) family. Geometry optimizations and electronic structure calculations are carried out in the framework of the local density functional theory using a planewave-pseudopotential scheme. The equilibrium cell geometry, the isothermal bulk modulus and its pressure derivatives have been evaluated for the two phases in a pressure range up to $400\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$. The Bader's Atoms in Molecules formalism is applied to characterize the chemical bonding and the atomic contributions to the bulk compressibility in $\ensuremath{\beta}$- and $\ensuremath{\gamma}\text{\ensuremath{-}}{A}_{3}{\mathrm{N}}_{4}$ crystals. The calculated stability diagram reveals the occurrence of a hypothetical $\ensuremath{\beta}\text{\ensuremath{-}}{\mathrm{C}}_{3}{\mathrm{N}}_{4}\ensuremath{\rightarrow}\ensuremath{\gamma}\text{\ensuremath{-}}{\mathrm{C}}_{3}{\mathrm{N}}_{4}$ phase transition around $370\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$. $\ensuremath{\Gamma}$-point harmonic vibrational modes are computed at selected pressures within the density functional perturbation theory approach. In agreement with recent theoretical calculations, the comparison of the computed Raman vibrational frequencies with experimental estimations for $\ensuremath{\beta}\text{\ensuremath{-}}{\mathrm{C}}_{3}{\mathrm{N}}_{4}$ raises doubts on a previously reported synthesis of this structure. Pressure effects on the vibrational frequencies inform of a reduction of the $P{6}_{3}∕m$ symmetry of the $\ensuremath{\beta}$ phase at pressures around $60\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$ and contribute to look into the mechanical stability of the $\ensuremath{\gamma}$ phase.