Continuum descriptions of epithelial tissue mechanics can replace expensive individual-based simulations with tractable macroscopic models, yet the link between cell-scale forces and tissue-scale transport remains poorly understood. We show that dimensionality controls this link: long-time mechanical relaxation rates reveal generalized porous-media-type nonlinear transport phenomena, D(ρ) ∝ ρ^γ. Exponents in nonlinear diffusivities are fixed by microscopic mechanics and dimensionality, providing a novel physical mechanism for emergent macroscopic transport phenomena.