  Variational solutions of the Boltzmann equation usually rely on the concept
of linear response. We extend the variational approach for tight-binding models
at high entropies to a regime far beyond linear response. We analyze both
weakly interacting fermions and incoherent bosons on a lattice. We consider a
case where the particles are driven by a constant force, leading to the
well-known Bloch oscillations, and we consider interactions that are weak
enough not to overdamp these oscillations. This regime is computationally
demanding and relevant for ultracold atoms in optical lattices. We derive a
simple theory in terms of coupled dynamic equations for the particle density,
energy density, current and heat current, allowing for analytic solutions. As
an application, we identify damping coefficients for Bloch oscillations in the
Hubbard model at weak interactions and compute them for a one-dimensional toy
model. We also approximately solve the long-time dynamics of a weakly
interacting, strongly Bloch-oscillating cloud of fermionic particles in a
tilted lattice, leading to a subdiffusive scaling exponent.
