Long, multimodal documents force retrieval-augmented systems to assemble answers from evidence fragmented across text, tables, and slides broken across cells in a long table, spread over multiple slides, or split between a figure and its discussion. Top-$k$ chunk retrieval treats each fragment independently and cannot represent how evidence connects. We introduce FLOWREADER, which reframes evidence assembly as a min-cost flow problem on a multimodal node graph: a single scoring vector $h$ controls source selection (via MMR), sink selection (via a length-aware answerability proxy), and the costs and capacities of every edge. The optimal flow is decomposed into candidate evidence paths, a compact non-redundant subset is selected by entropy-regularized replicator dynamics, and parallel VLM workers under a dual-process gate produce the answer with a single System-2 refinement pass triggered when answer consistency is low or the routed flow is strained. On VisDoMBench, FLOWREADER is best on the two subsets dominated by fragmented evidence PaperTab ($58.40$, $+1.30$ over G^{2}-Reader) and SlideVQA ($72.93$, $+0.62$) and competitive on SPIQA, FetaTab, and SciGraphQA. Macro-averaged across all five subsets, FLOWREADER ($65.47$) is within $0.74$ of the strongest baseline (G^{2}-Reader, $66.21$). Overall, these results show that min-cost flow performs well on fragmented multimodal evidence, where top-$k$ retrieval fails. It also provides a unified way to control scoring, routing, selection, and adaptive compute together.