This is already solved, but I decided it should be here on the forum, for future reference.

My system contains an MSI X79A-GD45 Plus motherboard, with a i7-4930k processor with 32GB RAM.
It has an NVIDIA RTX 1070 graphics card.

For capturing, I used to have the Blackmagic Intensity Pro as the system's capture card, but it caused lag when I tried to use it in conjunction with vmixcalls.

I also had a Yuan SC5104N 4 HDMI input card lying around. This capture card did not cause the aforementioned lag. But it would 'hang' a random points in time. And only fully restarting the computer would solve it.

I tried installing, reinstalling, uninstalling drivers. But the random hanging would persist.

Nothing indicated where the fault could lie.

And then I decided to look up the manual of my computer's motherboard.
It was very particular in which slots where PCI-E 3.0 and which were 2.0.

My graphics card, for whatever reason, occupied a 2.0 slot. And so did the capture card.
I switched them into a 3.0 slot and the card has been working smoothly for hours on end as we speak.

I'll still have to perform more endurance tests, but for now, it seems that that may have been the issue.