  We develop a framework inspired by Lauret's "bracket flow" to study the
generalized Ricci flow, as introduced by Streets, on discrete quotients of Lie
groups. As a first application, we establish global existence on solvmanifolds
in arbitrary dimensions, a result which is new even for the pluriclosed flow.
We also define a notion of generalized Ricci soliton on exact Courant
algebroids that is geometrically meaningful and allows for non-trivial
expanding examples. On nilmanifolds, we show that these solitons arise as
rescaled limits of the generalized Ricci flow, provided the initial metrics
have "harmonic torsion", and we classify them in low dimensions. Finally, we
provide a new formula for the generalized Ricci curvature of invariant
generalized metrics in terms of a moment map for the action of a non-reductive
real Lie group.
