Multiple Instance Learning (MIL) is the dominant framework for gigapixel whole-slide image (WSI) classification in computational pathology. However, current MIL aggregators route all instances through a shared pathway, constraining their capacity to specialise across the pathological heterogeneity inherent in each slide. Mixture-of-Experts (MoE) methods offer a natural remedy by partitioning instances across specialised expert subnetworks; yet unconstrained softmax routing may yield highly imbalanced utilisation, where one or a few experts absorb most routing mass, collapsing the mixture back to a near-single-pathway solution. To address these limitations, we propose ROAM (Region-graph OptimAl-transport Mixture-of-experts), a spatially aware MoE-MIL aggregator that routes region tokens to expert poolers via capacity-constrained entropic optimal transport, promoting balanced expert utilisation by construction. ROAM operates on spatial region tokens, obtained by compressing dense patch bags into spatially binned units that align routing with local tissue neighbourhoods and introduces two key mechanisms: (i) region-to-expert assignment formulated as entropic optimal transport (Sinkhorn) with explicit per slide capacity marginals, enforcing balanced expert utilisation without auxiliary load-balancing losses; and (ii) graph-regularised Sinkhorn iterations that diffuse routing assignments over the spatial region graph, encouraging neighbouring regions to coherently route to the same experts. Evaluated on four WSI benchmarks with frozen foundation-model patch embeddings, ROAM achieves performance competitive against strong MIL and MoE baselines, and on NSCLC generalisation (TCGA-CPTAC) reaches external AUC 0.845 +- 0.019.