Control of microtubule abundance, stability, and length is crucial to regulate intracellular transport as well as cell polarity and division. How microtubule stability depends on tubulin addition or removal at the dynamic ends is well studied. However, microtubule rescue, the event when a microtubule switches from shrinking to growing, occurs at tubulin exchange sites along the shaft. Molecular motors have recently been shown to promote such exchanges. Using a stochastic theoretical description, we study how microtubule stability and length depend on motor-induced tubulin exchange and thus rescue. Our theoretical description matches our in vitro experiments on microtubule dynamics in the presence of kinesin-1 molecular motors. Although the overall dynamics of a population of microtubules can be captured by an effective rescue rate, by assigning rescue to exchange sites, we reveal that the dynamics of individual microtubules within the population differ dramatically. Furthermore, we study in detail a transition from bounded to unbounded microtubule growth. Our results provide novel insights into how molecular motors imprint information of microtubule stability on the microtubule network.