We describe a mechanism to control the energy and magnetization currents in an artificial spin chain, consisting of an array of permalloy nanodisks coupled through a magnetodipolar interaction. The chain is kept out of equilibrium by two thermal baths with different temperatures connected to its ends, which control the current propagation. Transport is enhanced by applying a uniform radio-frequency pump field resonating with some of the spin-wave modes of the chain. Moreover, the two currents can be controlled independently by tuning the static field applied on the chain. Thus we describe two effective means for the independent control of coupled currents and the enhancement of thermal and spin-wave conductivity in a realistic magnonics device, suggesting that similar effects could be observed in a large class of nonlinear oscillating systems.