Culturing living cells in three-dimensional environments increases the biological relevance of laboratory experiments, but requires solutes to overcome a diffusion barrier to reach the centre of cellular constructs. We present a theoretical and numerical investigation that brings a mechanistic understanding of how microfluidic culture conditions, including chamber size, inlet fluid velocity and spatial confinement, affect solute distribution within three-dimensional cellular constructs. Contact with the chamber substrate reduces the maximally achievable construct radius by 15%. In practice, finite diffusion and convection kinetics in the microfluidic chamber further lower that limit. The benefits of external convection are greater if transport rates across diffusion-dominated areas are high. Those are omnipresent and include the diffusive boundary layer growing from the fluid-construct interface and regions near corners where fluid is recirculating. Such regions multiply the required convection to achieve a given solute penetration by up to 100, so chip designs ought to minimize them. Our results define conditions where complete solute transport into an avascular three-dimensional cell construct is achievable and applies to real chambers without needing to simulate their exact geometries.