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 depends on motor-induced tubulin exchange and thus rescue. Our theoretical description matches our in vitro experiments on microtubule dynamics in presence of kinesin-1 molecular motors. Although the average microtubule dynamics can be captured by an effective rescue rate, the dynamics of individual microtubules differs dramatically when rescue occurs only at exchange sites. 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.