A new paradigm for selective pumping of fluids in a complex network of channels in the microscale flow regime is presented. The model is inspired by internal flow distributions produced by the rhythmic wall contractions observed in many insect tracheal networks. The approach presented here is a natural extension of previous two-dimensional modeling of insect-inspired microscale flow transport in a single channel, and aims to manipulate fluids efficiently in microscale networks without the use of any mechanical valves. This selective pumping approach enables fluids to be transported, controlled and precisely directed into a specific branch in a network while avoiding other possible routes. In order to present a quantitative analysis of the selective pumping approach presented here, the velocity and pressure fields and the time-averaged net flow that are induced by prescribed wall contractions are calculated numerically using the method of fundamental solutions. More specifically, the Stokeslets-meshfree method is used in this study to solve the Stokes equations that govern the flow motions in a network with moving wall contractions. The results presented here might help in understanding some features of the insect respiratory system function and guide efforts to fabricate novel microfluidic devices for flow transport and mixing, and targeted drug delivery applications.