I have had Wii U and Vita since their respective launch and got PS4 at launch as well... so I tried Vita for Remote Play
It has two ways of connecting. One is via your wireless router (which gives more range than Gamepad) and other is directly to PS4 which has same range as Gamepad. The connect directly has significantly less input lag than the wireless router option.
I was able to play Resogun on Master when connected directly to PS4 which means its pretty damn responsive even if there is some miniscule lag. I couldn't play it on Master when connecting via wireless (dying way too fast). When connected directly, there is probably some miniscule lag, but I couldn't really tell. For me the bigger issue was the controls don't have parity. Needing to touch a zone on the rear-touch for L2/R2 or L3/R3 are the bigger issues than amount of lag.