Members are being charged early
In the screenshot above -&gt; I don't see any double charges. Every Contribution in the screenshot belongs to a different recurring series (the id = numbers are different).
As far as at the same time on the same day -&gt; if cron has not run for some time on your site -&gt; then this is exactly what you would see. Every recurring series that has a next_scheduled_date in the past would be due as soon as iATS Payments Recurring Contributions (Daily) is triggered by your system cron.
As far as double charges -&gt; there are safe guards in place for that -&gt; it can really only happen if that cron job is triggered by two crons at the same time and/or if you have a production and a development set up and you have cron running on both.
PS - Make sure you are on the latest iATS Extension update (1.7.4)
On timing -&gt; which recurring contribution series should create a new contribution  to be submitted for processing to iATS -&gt; is solely governed by the Next Scheduled Contribution Date in the Recurring Series Record.
In this report -&gt; /civicrm/report/com.iatspayments.com/recur?reset=1 you can filter Next Scheduled Contribution Date -&gt; next week so see what's going to happen next week. For this client -&gt; they will have two series transact on April 09 and one on April 10.

It sounds like your recurring series definitions/dates are not in line with your membership dates.
Also very important - do not switch ON Recurring within your iATSpayments.com account itself as then you have both civicrm and iats trying to control the schedule of your payments.
So go into one of your recurring series records (for example id = 578 you have in your second screenshot) -&gt; and view all variables re: time -&gt; i.e. next scheduled date + frequency