Recent theories suggest planetesimal formation via streaming and/or gravitational instabilities may be triggered by localized enhancements in the dust-to-gas ratio, and one hypothesis is that sufficient enhancements may be produced in the pile-up of small solid particles inspiralling under aerodynamic drag from the large mass reservoir in the outer disc. Studies of particle pile-up in static gas discs have provided partial support for this hypothesis. Here, we study the radial and temporal evolution of the dust-to-gas ratio in turbulent discs that evolve under the action of viscosity and photoevaporation. We find that particle pile-ups do not generically occur within evolving discs, particularly if the introduction of large grains is restricted to the inner, dense regions of a disc. Instead, radial drift results in depletion of solids from the outer disc, while the inner disc maintains a dust-to-gas ratio that is within a factor of ∼2 of the initial value. We attribute this result to the short time-scales for turbulent diffusion and radial advection (with the mean gas flow) in the inner disc. We show that the qualitative evolution of the dust-to-gas ratio depends only weakly upon the parameters of the disc model (the disc mass, size, viscosity and value of the Schmidt number), and discuss the implications for planetesimal formation via collective instabilities. Our results suggest that in discs where there is a significant level of midplane turbulence and accretion, planetesimal formation would need to be possible in the absence of large-scale enhancements. Instead, trapping and concentration of particles within local turbulent structures may be required as a first stage of planetesimal formation.