Two-dimensional collective cell migration assays are used to study malignant spreading and tissue repair. These assays involve combined cell migration and cell proliferation processes, both of which are modulated by cell-to-cell crowding effects. Previous discrete models of two-dimensional collective cell migration assays involve a nearest-neighbour proliferation mechanism where crowding effects are incorporated by aborting potential proliferation events if the randomly chosen target site is occupied. There are two potential limitations of this traditional approach: (i) it seems unreasonable to abort a potential proliferation event based on the occupancy of a single, randomly chosen target site; and, (ii) the continuum limit description of this mechanism leads to the standard logistic growth function, but there is increasing evidence suggesting that cells do not always grow logistically. Motivated by these limitations we introduce a generalised proliferation mechanism into a two-dimensional lattice-based exclusion process model, which allows non-nearest neighbour proliferation events to take place over a template of <em>r</em> ≥ 1 concentric rings of lattice sites. Further, the decision to abort potential proliferation events is made using a <em>crowding function</em>, <em>f</em>(<em>C</em>) ∈ [0,1] with <em>f</em>(0) = 1 and <em>f</em>(1) = 0. This approach accounts for the density of agents within a group of lattice sites rather than dealing with the occupancy of a single site only. Analysing the continuum limit description of the stochastic model shows that the standard logistic source term, λ<em>C</em> (1 − <em>C</em>), where λ is the proliferation rate, is generalised to a universal growth function, <em>λ</em>C<em>f</em>(<em>C</em>). Comparing the solution of the continuum description with averaged simulation data indicates that the continuum model performs well for many choices of <em>f</em>(<em>C</em>) and <em>r</em>. For nonlinear <em>f</em>(<em>C</em>), the quality of the continuum-discrete match increases with <em>r</em>. Therefore, we suggest that estimating <em>r</em> from time lapse images will help distinguish between situations where the simpler continuum model is adequate from other situations where repeated simulations of the stochastic algorithm is required.