Normally people will try to upgrade either at the time of booking (if they bought a ridiculously expensive Latitude fare) or when their booking window opens - depends on status 7/4/2 days prior to departure). You can also request an upgrade when you check in on-line or at a check-in kiosk (and presumably in person). Any of these will "cost" you eUpgrades.

If there is R space available, you get the upgrade right away - otherwise they give it to you just prior to boarding (sometimes after boarding has started) - or not if the J section checks in full or others are higher up on the waitlist. This is called Aerolotto

Occasionally people will be given a "free" upgrade - often called Op Ups (Operational Upgrades). These happen when there are seats available in J but the back has been oversold. They will generally upgrade people on the eUpgrade list first (and take their eUpgrades) before offering a free Op Up - which is supposed to be done by status. Occasionally harried gate staff hand them out randomnly just to get the flight out, but that is not supposed to happen.