Malicious entities abuse advanced modulation classification (MC) techniques to launch traffic analysis, selective jamming, evasion, and poison attacks. Recent studies show that current defenses against such attacks are static in nature and vulnerable to persistent adversaries who invest time and resources into learning the defenses, thereby being able to design and execute more sophisticated attacks to circumvent them. In this paper, we present a moving-target defense framework to support a novel modulation-masking mechanism we develop against advanced and persistent MC attacks. The modulated symbols are first masked using small perturbations to make them appear to an adversary in a state of ambiguity about the model as if they are from another modulation scheme. By deploying a pool of deep learning models and perturbation-generating techniques, our defense strategy keeps changing (moving) them as needed, making it difficult (cubic time complexity) for adversaries to keep up with the evolving defense system over time. We show that the overall system performance remains unaffected under our technique. We further demonstrate that, over time, a persistent adversary can learn and eventually circumvent our masking technique, along with other existing defenses, unless a moving target defense approach is adopted.