Lymph transport inside lymphatic vessels is highly complex and not yet fully understood. So far, a consensus has not been reached among existing analytical models on how spatiotemporal coordination of contracting adjacent lymphangions affects lymph transport. To understand complex lymph transport, we created a novel microfluidic valvular chip with flexible bicuspid valves and segmental pneumatic pumps based on a microfluidic device with an inside 3D structure made of hydrogels. Inside the chip, water moved unidirectionally when the microfluidic channel was locally compressed, with its velocity profile closely resembling the waveform of lymph observed in vivo. Furthermore, for a systematic and mechanistic study, we constructed a numerical model based on fluid–structure interaction and validated the model via demonstration of similarities in water transport characteristics between the model and the chip. Using this model, we examined various mechanical and time-dependent parameters, such as period, phase delay, sequence, and strength of contractions, valve compliance, fluid viscosity, and pressure differences, for their effects on water transport. Although our model is simplified, it enabled a parametric study that helped clarify the mechano-temporal correlations between compressions of adjacent chambers via transmissions of hydrodynamic forces, which regulate complex lymph transport. Moreover, our chip demonstrated technical advances that enable unidirectional discrete movement of fluid in the picoliter range by phenumatic pumping. The velocity profile is also similar to the pulse waveform of arteries under pathological conditions such as increased aortic stiffness, allowing our chip to be used for in vitro mechanobiology studies of endothelial cells.