Subscription not yet started
Navigate to the subscription you wish to change and select the pencil icon beside 'Start'. You can then select any future date.
Select the new date and press Preview changes and then confirm when you are happy by pressing Change start date.
That's all there is to it. Move in info will now be sent to the customer on the new date as well as payments being collected from this new date.
Subscription already started
This is useful if the customer has advised they cannot move in after the first invoice has already been taken. You can change the move in date to the desired date, and the next invoice will be taken a month/4weeks after that. This may save you having to refund and recreate the subscription.
Use the pencil icon as above to edit the date.
In Stripe you can then check the upcoming invoice.