It does not quite sound like what you're describing but make your own judgement.
Screen tearing is normally associated as a mismatch between the video signal and the refresh rate of the display. Again it is most associated with high end gaming whereby the video signal due to the complexity of the images being rendered cannot keep up with the refresh rate of the display.
Various methods have been devised to help automatically compensate for this, the three main ones I am aware of are V-Sync, G-Sync and FreeSync.
Unfortunately Apple -
- Have historically been backwards regarding video drivers
- Have historically been poor at supporting the (Mac) gaming community
As a result as far as I am aware they still do not support Freesync which would work with various AMD video cards and for other reasons also not supported G-Sync which works with Nvidia cards. I have no idea about V-Sync support from Apple.
It is also the case that some monitors only support one of the above standards so if you are concerned about this you need to ensure the monitor matches your video card and driver. (Likely to be a pointless exercise with Macs.)
See - https://en.wikipedia.org/wiki/Screen_tearing
Unless you were that extremely rare creature - a Mac gamer I would not expect this normally to show up. However if you were doing heavy 4K video processing and playback maybe then your antique Mac Pro ;) might exhibit this. Of course another problem is that the video card in your antique Mac Pro cannot be upgraded so it is likely an older model which does not support Freesync with no option to fit a newer one which would e.g. an RX 580 even if Apple had driver support for this feature - which they don't.
I would certainly take in to consideration Grant's suggestion about checking which Thunderbolt bus you are using. See this Apple article on that topic Use multiple displays with your Mac Pro (Late 2013) - Apple Support
It would not be beyond the realms of possibility that this is instead a symptom of a hardware fault or as has happened in the past a manufacturing fault in the GPU. If you feel it is not the Thunderbolt issue or a performance issue then it maybe worth asking Apple to do testing on it. At least your 2013 model is easier to transport than both the classic Mac Pro range and the forthcoming Mac Pro 2019.
PS. The only way to do 4K over HDMI at 60Hz is to get an 'Active' Displayport to HDMI adapter like this one https://www.club-3d.com/en/detail/2339/mini_displayport_1.2_to_hdmi_2.0_active_adapter/