We present a hybrid lattice Boltzmann algorithm for the simulation of flow glass-forming fluids, characterized by slow structural relaxation, at the level of the Navier-Stokes equation. The fluid is described in terms of a nonlinear integral constitutive equation, relating the stress tensor locally to the history of flow. As an application, we present results for an integral nonlinear Maxwell model that combines the effects of (linear) viscoelasticity and (nonlinear) shear thinning. We discuss the transient dynamics of velocities, shear stresses, and normal stress differences in planar pressure-driven channel flow, after switching on (startup) and off (cessation) of the driving pressure. This transient dynamics depends nontrivially on the channel width due to an interplay between hydrodynamic momentum diffusion and slow structural relaxation.