The structure of the equations of motion of a time-dependent mechanical system, subject to time-dependent non-holonomic constraints, is investigated in the Lagrangian as well as in the Hamiltonian setting. The treatment applies to systems with general nonlinear constraints, and the ambient space in which the constraint submanifold is embedded is equipped with a cosymplectic structure. In analogy with the autonomous case, it is shown that one can define an almost-Poisson structure on the constraint submanifold, which plays a prominent role in the description of non-holonomic dynamics. Moreover, it is seen that the corresponding almost-Poisson bracket can also be interpreted as a Dirac-type bracket. Systems with a Lagrangian of mechanical type and affine non-holonomic constraints are treated as a special case and two examples are discussed.