Flexible DNA molecules are imaged by fluorescence microscopy while undergoing convection through micron-sized pores of a packed sphere bed. To circumvent the high pressures of a closed channel design, an open channel microfluidic device is selected such that sphere wicking drives convection. Image analysis yields the average level of chain stretching and the average DNA migration rate as a function of Deborah number De for ratios of sphere diameter to coil radius of gyration of order 1 to 10. Visualizations also reveal the transient localized conformations and trajectories of individual molecules, which can migrate only by traversing coil-sized pore constrictions. Two De (flow) transitions are noted: the first, at De similar to 0.3, leads to limited chain stretching, while the second, at De similar to 1, leads to gross chain stretching via a mechanism of entanglement about sphere contact points. Entangled chains adopt elongated "U"- and "J"-shaped conformations, which are relaxed by "pulley-like" sliding over entanglement points. Entanglement onset is associated with increase in average DNA migration rate, which abruptly shifts from below to above that for solvent. Results are discussed in the context of visualization results from gel electrophoresis and implications for polymer chromatography.