Machines made of soft materials bridge life sciences and engineering. Advances in soft materials have led to skin-like sensors and muscle-like actuators for soft robots and wearable devices. Flexible or stretchable counterparts of most key mechatronic components have been developed, principally using fluidically driven systems6–8; other reported mechanisms include electrostatic, stimuli-responsive gels and thermally responsive materials such as liquid metals and shape-memory polymers. Despite the widespread use of fluidic actuation, there have been few soft counterparts of pumps or compressors, limiting the portability and autonomy of soft machines. Here we describe a class of soft-matter bidirectional pumps based on charge-injection electrohydrodynamics. These solid-state pumps are flexible, stretchable, modular, scalable, quiet and rapid. By integrating the pump into a glove, we demonstrate wearable active thermal management. Embedding the pump in an inflatable structure produces a self-contained fluidic ‘muscle’. The stretchable pumps have potential uses in wearable laboratory-on-a-chip and microfluidic sensors, thermally active clothing and autonomous soft robots.