Streams and rivers are important components of the carbon cycle as they transport and transform dissolved organic matter (DOM). Using high‐resolution Fourier‐transform ion cyclotron resonance mass spectrometry, we studied the spatial distribution of DOM at the molecular level at more than 100 sites across a stream network during summer and winter baseflow. We developed a model approximating the time DOM spent in the fluvial network, a key constraint on the biogeochemical processing of DOM. Discharge‐weighted travel time explained the compositional changes of DOM, which differed markedly in summer and winter. We attribute these seasonal differences to variation in source material, putatively reflecting the dynamics of freshly produced DOM in summer and DOM with an imprint of leaf litter in winter. Hydrological mixing was an important driver of the spatial dynamics of DOM. From the convergence rate of DOM compound intensities to the network‐wide average, we inferred the spatial distribution of sources within the catchment. Finally, we estimated network‐wide apparent mass transfer coefficients (vf app) of individual DOM compounds, which describe the vertical velocity at which DOM compounds are removed by biotic and abiotic processes. We identified the oxidative state of carbon as an important factor explaining vf app, which we consequently attribute to biological uptake of thermodynamically favorable DOM compounds. This work contributes to our understanding of the spatial processes, temporal constraints, and chemical properties of DOM that regulate the transformation and diagenesis of DOM at the fluvial network scale.