Many times, I’ve been asked about the right time to release/send references during an interview process. Some employers ask/require them right away, others after the candidate has had an in-person interview and is a finalist for a position.
The advice I would offer is to give the employer your references AFTER an in-person interview has taken place and you have been given indications that you are a finalist for a position. I don’t feel it is appropriate for employers to ask for references before they have even met you or after an initial phone interview. They have not fully vetted the you NOR have you evaluated the company and position fully. In addition, asking your references to write one multiple times for too many positions wears thin.
Companies using applicant tracking systems tend to ask for reference write-ups early in the process. Unfortunately, these systems have no flexibility in the sequence. If you are asked for references early in the process, ask the appropriate HR contact about releasing your references later in the process, for the reasons previously mentioned.