Forced stratified flows are shown to suffer two types of linear long-wave instability: a ‘viscous’ instability which is related to the classical instability of Kolmogorov flow, and a ‘conductive instability’, with the form of a large-scale, negative thermal diffusion. The nonlinear dynamics of both instabilities is explored with weakly nonlinear theory and numerical computations. The introduction of stratification suppresses the viscous instability, but also makes it subcritical. The second instability arises with stronger stratification and creates a prominent staircase in the buoyancy field; the steps of the staircase evolve over long timescales by approaching one another, colliding and merging (coarsening the staircase).