While viewing an issue via the standard JIRA page I would like a control next to "View on Rapid Board" that allows me to send the issue to any sprint that has not yet been completed on any Rapid Board on which the issue currently appears.

While allowing issues to be assigned to sprints that have already started in addition to sprints that have not yet started is not good scrum practice, it happens in reality and the control should not restrict the user from doing so, (though it could be useful to indicate that the sprint is already in progress.)