Often1

Sort of. But, remember that Downgrades are refunds and not compensation. If your flight is cancelled and you are then rebooked onto a subsequent service, but in a lower class, without getting into the vagaries of why the flight was cancelled, the passenger would have a cancellation compensation claim, the amount determined by distance (Type I, II or III). In addition, the passenger would be entitled to a refund of 75% of the ticket's segment cost.

This is, however, entirely different from the situation where the second flight is also cancelled. That is because EC 261/2004 bases compensation on a ticket and it looks only at the arrival time at the "final ticketed destination". Thus, whether you are ticketed on one flight which arrives 5 hours late or sequentially on 2 or 3 flights each are cancelled and you arrive 10 or 20 hours late, you are still only entitled to the compensation for the number of hours you are delayed at that destination.

Take the hypothetical of a passenger with a single ticket LHR-JFK. His flight is cancelled and he is rebooked for the same flight the next day. That flight arrives at JFK 5 hours late. Thus, the passenger arrives at his final ticketed destination, e.g. JFK, 31 hours late. Under EC 261/2004, the compensation for a flight of that distance is EUR 600. It is not EUR 1,200 because he suffered two delay/cancellations, but rather one delay of 31 hours at his final ticketed destination on one ticket.