  The Yang-Mills gradient flow for QCD-like theories is generalized by
including a fermionic matter term in the gauge field flow equation. We combine
this with two different flow equations for the fermionic degrees of freedom.
The solutions for the different gradient flow setups are used in the
perturbative computations of the vacuum expectation value of the Yang-Mills
Lagrangian density and the field renormalization factor of the evolved fermions
up to next-to-leading order in the coupling. We find a one-parameter family of
flow systems for which there exists a renormalization scheme in which the
evolved fermion anomalous dimension vanishes to all orders in perturbation
theory. The fermion number dependence of different flows is studied and
applications to lattice studies are anticipated.
