A system for moving a quantity of liquid from an origin or source cavity to a destination cavity in a network of microfluidic cavities, wherein the origin or source cavity and the destination cavity are separated by a valveless microfluidic channel having a resistance to fluid flow greater than that of the source cavity, the system comprising: a receptacle for receiving and engaging with a flowchip comprising the network of microfluidic cavities; and a pressure sequencer comprising a set of gas valves and configured to be connected to a first gas source for producing a high gas pressure in microfluidic cavities, a second gas source for producing a low pressure in microfluidic cavities, and a third gas source for producing an intermediate gas pressure in microfluidic cavities, wherein the high gas pressure is greater than the low pressure, the intermediate gas pressure is less than the high gas pressure but greater than the low pressure, and the intermediate gas pressure is insufficiently great overcome resistance to fluid flow in the microfluidic channel when the source cavity is substantially empty of the liquid, wherein the pressure sequencer can apply any pressure state to any cavity; characterized in that the system further comprises, a controller configured to direct the pressure sequencer to: (a) apply the high gas pressure to any cavity (other than the destination cavity) connected to the origin or source cavity by a first channel, while applying the low gas pressure to any cavity (other than the origin cavity) connected to the destination cavity by a second channel, to move a portion of the quantity of liquid from the origin or source cavity, through the microfluidic channel, and to the destination cavity, and (b) apply an intermediate gas pressure to the origin or source cavity by the first channel before the quantity of liquid is completely removed from the source cavity, wherein the intermediate gas pressure is sufficiently great to push at least some of the quantity of liquid remaining after (a) to the destination cavity, but avoids introducing gas into the microfluidic channel. a receptacle for receiving and engaging with a flowchip comprising the network of microfluidic cavities; and a pressure sequencer comprising a set of gas valves and configured to be connected to a first gas source for producing a high gas pressure in microfluidic cavities, a second gas source for producing a low pressure in microfluidic cavities, and a third gas source for producing an intermediate gas pressure in microfluidic cavities, wherein the high gas pressure is greater than the low pressure, the intermediate gas pressure is less than the high gas pressure but greater than the low pressure, and the intermediate gas pressure is insufficiently great overcome resistance to fluid flow in the microfluidic channel when the source cavity is substantially empty of the liquid, wherein the pressure sequencer can apply any pressure state to any cavity; characterized in that the system further comprises, a controller configured to direct the pressure sequencer to: (a) apply the high gas pressure to any cavity (other than the destination cavity) connected to the origin or source cavity by a first channel, while applying the low gas pressure to any cavity (other than the origin cavity) connected to the destination cavity by a second channel, to move a portion of the quantity of liquid from the origin or source cavity, through the microfluidic channel, and to the destination cavity, and (b) apply an intermediate gas pressure to the origin or source cavity by the first channel before the quantity of liquid is completely removed from the source cavity, wherein the intermediate gas pressure is sufficiently great to push at least some of the quantity of liquid remaining after (a) to the destination cavity, but avoids introducing gas into the microfluidic channel. (a) apply the high gas pressure to any cavity (other than the destination cavity) connected to the origin or source cavity by a first channel, while applying the low gas pressure to any cavity (other than the origin cavity) connected to the destination cavity by a second channel, to move a portion of the quantity of liquid from the origin or source cavity, through the microfluidic channel, and to the destination cavity, and (b) apply an intermediate gas pressure to the origin or source cavity by the first channel before the quantity of liquid is completely removed from the source cavity, wherein the intermediate gas pressure is sufficiently great to push at least some of the quantity of liquid remaining after (a) to the destination cavity, but avoids introducing gas into the microfluidic channel.