Based on the first-principles calculations, the effects of pressure on the structural, magnetic, lattice dynamical, elastic and thermal expansion properties of TMFe3N (TM = Ni, Pd, Pt) compound are studied, and a comparison with the parent phase Fe4N is performed. The calculated total enthalpy reproduces that the Ni, Pd and Pt atoms favor staying in the Fe 1a site of parent phase, and forming the ferromagnetic state. The substitution of Ni, Pd and Pt atoms reduces the critical pressure of ferromagnetic collapse of the parent phase, while enhancing its dynamic stability when subjected to high pressure. The results of the elastic constants indicate that substituting Ni, Pd and Pt atoms can improve the mechanical characteristics of the parent phase. Pressure-induced elastic softening and negative thermal expansion are found in both Fe4N and TMFe3N (TM = Ni, Pd, Pt) compounds. These two abnormal behaviors are analyzed with the magnetoelastic force and the competition among the spontaneous volume magnetostriction effect, pressure-induced effect, and thermal effect within the system, respectively.