How do you handle user experience design within Scrum?

I've found that it works best when the UX designers are part of the Scrum Team and closely aligned with the Product Owner, as opposed to the Development Team.

The work of the UX designer tends to align with part of responsibilities of the Product Owner, two in particular - clearly expressing the Product Backlog Items and ensuring that the Development Team understands the Product Backlog items to the level needed. The user experience desired by a story is required for the Development Team to be able to provide a meaningful estimate to the Product Backlog Items, but the work doesn't end there. When work is being done, there's collaboration between the UX designer and the members of the Development Team to discuss any issues that arise when realizing the UX design and come up with solutions.

Because much of the work of the UX designer is forward-looking, it doesn't make much sense to put them on the Development Team unless the UX designer is also implementing the designs. As the Product Owner is representing the various stakeholders by capturing needs as Product Backlog Items and ordering them, the UX designer is working on understanding how those needs are best designed into the system. UX designers are also maintaining the design system (if you have one).

That said, a UX designer or UX design team could very easily be operating on a Sprint cadence, working from a backlog of UX design work, holding Daily Scrums to plan their day around Sprint Goals, reviewing their work with the Scrum Team(s) of an organization at the end of a Sprint, delivering UX designs or design system updates at least once a Sprint, at so on. The level of ceremony would depend on the size and complexity of the UX design work. However, even with a UX design team, I would recommend consistent and close partnerships with UX designers and Scrum Teams to the point of ensuring that every Scrum Team works closely with a UX designer (one UX designer may support multiple Scrum Teams).

This approach has risks as still it is creating dependency on outer teams to create a  'Done' increment.

How clear is that risk, and would it be acceptable to the Development Team? Are they confident that the management of that dependency lies under their control?

In Scrum, team members must be able to commit to each Sprint Goal. They cannot make commitments on other people’s behalf, and hence must be confident they have what it takes as a team to create a Done increment.