Agile is clearly expanding its horizon, digital is the age of customer, user or customer experience design becomes significant tasks for any customer-centric business, so how to apply Agile for managing the UX process itself, or to reframe the question: "How do you integrate UX processes and skills into a Agile iterative design and build cycle?"
It's applying some Agile concepts to the management of UX teams. To standardize on a management methodology and drive various UX teams; tasks, schedules, budgets and progress still need to be tracked somehow. Team collaboration and communication needs to be implemented somehow as well. If UX team can apply Agile and other methodologies to create an approach that helps teams and stakeholders meet their needs, in time and on budget, then it is win-win!
The agile communication method works fine in UX process: Development teams often run into fuzzy periods, as they don't have all the requirements or what they need to design the software architecture. UX process is really "design" work since they are trying to solve a new problem, and they have to come up with creative ideas to address the challenge. The outcome is not known and there's a lot of brainstorming going on. The agile approach can enforce interactive communication, prioritized backlog, daily stand ups, design process is already collaborative and iterative, and make incremental project improvement in UX/CX design to increase overall project success rate.
The most difficult part of UX design and Agile is where to fit concept into the process. Deliverables, such as wireframes, design flats and specs would all work in agile, but where does concept - white-boarding, research, and comparative analysis - fit into the process? It's often difficult to time-box - you don't want to rush the right solution, but you also need to know when to move on. The question becomes: Can agile cope with repeated iterations of research and requirements maturation, or concept modeling with no progressing past that point until it's digested all the needs and requirements and drafted an end-to-end experience model that works for users, the business and IT.
UX is NOT UI, it is much more than "design" and more often than not requires discipline, rigor, deep thought and a systematic approach to deliver success. Thus, Agile could be a nice fit philosophy and methodology upon focusing on how to adapt UX teams to work on Agile and integrate UX processes and skills into an iterative design and build cycle.