When miniaturizing fluidic circuitry, the solid walls of the fluid channels become increasingly important because they limit the flow rates achievable for a given pressure drop, and they are prone to fouling. Approaches for reducing the wall interactions include hydrophobic coatings, liquid-infused porous surfaces, nanoparticle surfactant jamming, changes to surface electronic structure, electrowetting, surface tension pinning and use of atomically flat channels. A better solution may be to avoid the solid walls altogether. Droplet microfluidics and sheath flow achieve this but require continuous flow of the central liquid and the surrounding liquid. Here we demonstrate an approach in which aqueous liquid channels are surrounded by an immiscible magnetic liquid, both of which are stabilized by a quadrupolar magnetic field. This creates self-healing, non-clogging, anti-fouling and near-frictionless liquid-in-liquid fluidic channels. Manipulation of the field provides flow control, such as valving, splitting, merging and pumping. The latter is achieved by moving permanent magnets that have no physical contact with the liquid channel. We show that this magnetostaltic pumping method can be used to transport whole human blood with very little damage due to shear forces. Haemolysis (rupture of blood cells) is reduced by an order of magnitude compared with traditional peristaltic pumping, in which blood is mechanically squeezed through a plastic tube. Our liquid-in-liquid approach provides new ways to transport delicate liquids, particularly when scaling channels down to the micrometre scale, with no need for high pressures, and could also be used for microfluidic circuitry.