The upgrade is a two step process - first we bring your system up to date with the latest jessie version of OSMC which will be a normal size update, at the end of this we modify /etc/apt/sources.list to point to the stretch repos.
The system then reboots and automatically triggers another update check to do the migration to stretch. This can be anywhere between 300-500 packages to upgrade. Yesterday (after most of you tested) we added an additional disk space check for this second phase download which will refuse to proceed if you have less than 300MB free on the root filesystem and 30MB free on /boot. (applicable to Pi’s) This is to avoid running out of user (non root reserved) space during the update which may cause Kodi to crash. The first stage update via My OSMC already had this disk space check in place.
If enough space is not available it will display a warning and then reboot back into Kodi. The system will be perfectly usable but you will still effectively be stuck on Jessie until you free up space and check for updates again, then the migration will proceed.