We analyze fermionic criticality in relativistic <math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> dimensional fermion systems using the functional renormalization group (FRG), concentrating on the Gross-Neveu (chiral Ising) and the Thirring model. While a variety of methods, including the FRG, appear to reach quantitative consensus for the critical regime of the Gross-Neveu model, the situation seems more diverse for the Thirring model with different methods yielding vastly different results. We present a first exploratory FRG study of such fermion systems including momentum-dependent couplings using pseudo-spectral methods. Our results corroborate the stability of results in Gross-Neveu-type universality classes, but indicate that momentum dependencies become more important in Thirring-type models for small flavor numbers. For larger flavor numbers, we confirm the existence of a non-Gaussian fixed point and thus a physical continuum limit. In the large-<math><mrow><mi>N</mi></mrow></math> limit, we obtain an analytic solution for the momentum dependence of the fixed-point vertex.