Hi @Peter Jackison and welcome to the Community!
There's quite a bit unclear to me about your use case here. From your description, it seems as if you are running a single sprint across multiple teams. On top of that, you seem to be taking the responsibility for running the team's work out of their hands - apart from a tooling question, I read lack of communication and ownership about ongoing work between the lines. But that may be just me taking assumptions to fill in the blanks ...
In an ideal situation, you would have a sprint per team, with each team being responsible for planning its own sprints. In an ART, these sprints should have the same start and end dates, so it becomes possible to keep aligned across the teams in your ART. When each team's velocity becomes predictable, the idea should be not to be moving open items to the next sprint all the time, but try not to overcommit. If all teams are working in the same sprint, you won't be able to get an idea of each individual team's velocity and their capacity to deliver. Maybe this one team is currently not updating the status of their work in real time, which might mean that the open items you move to another sprint are not really open. If that is the case, you should have a discussion about that with them.
Also, Jira helps you to move unfinished work to the next sprint automatically when you complete the sprint. It is not something you should do beforehand as indeed, items that could be finished at the very end of the sprint may disappear from view this way, while they are relevant for the team currently working on them.
Overall and in summary: the key here is all about communication. Rather than moving items around, talk to each other, find out where you can improve and make agreements on that. Together. As RTE, help people become autonomous while helping them see the bigger picture they contribute to.
Hope this helps!
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