Mathematical solutions are obtained for several sedimentary problems featuring semi-infinite alluvial channels evolving under diffusional sediment transport. Moving boundaries are considered at one end of the channels, and represent transitions between alluvial reaches and either bedrock-floored channels or bodies of standing water. Three elementary cases are considered: bedrock–alluvial transitions, lake breaches, and prograding deltas. It is shown that idealized formulations of all three problems share the same mathematical structure and admit exact similarity solutions. Elementary solutions can further be assembled to describe composite profiles. This is illustrated by the case of a natural lake undergoing simultaneous breaching and backfill. For both elementary and composite cases, the explicit solutions clarify the link between alluvial profile evolution and the migration of channel boundaries. For the case of lake breaching, for instance, the outlet channel profile is controlled simultaneously by downwards incision and upstream migration of the channel head. The pace of the resulting water-level drawdown in turn affects the form of the backfill deposits upstream of the lake.