Blackmagic's Linux drivers are gratis, but not libre. And of course they aren't upstream. This means that they are only guaranteed to work if you use exactly the same kernel that Blackmagic supports -- we've had all sorts of problems with them: the cards lock up (requiring a reboot before they will work again); the whole system locks up; the drivers were randomly overwriting other processes' memory (!); etc.

However our reliability requirements may be different than yours & the OP's, as we (at http://stb-tester.com ) use video capture for automated UI testing so we need rock-solid reliability.

I concur with the other commenter that devices with an upstream driver --such as the UVC driver-- are the way to go.