Can I send surveys to candidates when they move to the next stage of the recruitment process?

No, it's not possible to trigger a Starred survey based on stage change or on the submission of an interview scorecard.

The survey triggers that are available in Starred are based on the candidate leaving your recruiting pipeline, which means either being rejected, hired or having withdrawn.

The reason why you can't trigger surveys to candidates based on a stage change or the submission of a scorecard from an interviewer in your Applicant Tracking System is that the candidate receiving the survey would still be in the pipeline for the job and therefore might not give reliable feedback. They might in fact be biased and give a more positive feedback in hopes of being hired, which leads to skewed survey results.