Final answer:
The AFL-CIO is in charge of sustaining and improving the online learning management systems for training centers across the nation, fostering continual education and skill enhancement.
Step-by-step explanation:
The joint labor/management-sponsored organization responsible for maintaining and upgrading the multiple nationwide online learning management systems utilized by training centers is the AFL-CIO. The AFL-CIO, specifically through its Training and Apprenticeships program, collaborates with various training centers to ensure that their online learning platforms are state-of-the-art and effective for worker education and skill development. This cooperation exemplifies the commitment of both management and labor to continuous learning and professional growth within various industries.