You can play PlayStation 5 games with a DualShock controller on a PlayStation 4 via Remote Play.
Eurogamer has confirmed the feature works. There’s even a menu, accessed by pressing the PlayStation Button, that lets you jump to the PS5 Control Centre, among other things. However, you are not able to use a DualSense controller with the PS5 Remote Play app on PS4.
The PS5 Remote Play app popped up on PS4 yesterday. The app lets you use the PS4 to connect to and control your PS5. The options let you switch between 540p, 720p standard and 1080p streaming.