From the Adventurers rulebook:
The -2 cost token also goes on an Action Supply pile. It reduces the cost of cards from that pile on a particular player's turn. The -2 cost token lowers costs for all purposes; for example with a -2
cost token on Gear, a player could Transmogrify Copper into Gear. The -2 cost token does not lower costs below 0. This token is placed with Ferry.
From the Dark Ages rulebook:
For example, if you use Procession to play Band of Misfits twice and choose Fortress the first time, you will automatically replay it as Fortress, then trash the Band of Misfits, return it to your hand (it is a Fortress when it's trashed, and Fortress has a when-trashed ability that returns it to your hand), and gain an Action card costing exactly 6 (1 more than Band of Misfits, which has left play and so is no longer copying Fortress).
It would appear ritual uses the cost of the card at the time it was trashed but procession uses the cost of the card when it is in the trash. I cannot immediately see where the timing makes a difference for ritual however the timing is important for procession.