There are a lot of figures about lag time in milliseconds between different types of cameras, and defense of EVFs by those who use them, and counter-arguments for OVFs by those who use them. I can't say whose figures, if any, are correct, or honestly whether any of those figures really address your question.
So here's how I'll try to put it, as someone who shoots lots of bird-in-flight shots with two different cameras, one a DSLR with an OVF and the other a mirrorless with an EVF.
First, there's not a black-and-white, one can do it and the other can't, answer as many want to argue there is. BOTH types can be used for action/movement/panning. Leaving out other factors, and assuming focus systems are equally capable (the bigger issue with many mirrorless cameras is the ability, or lack thereof, to track focus continually on a fast-moving target...some are better than others but few if any approach DSLRs)...there is a very slight lag in an EVF that likely would not affect most shooters capturing single-shot action scenes, or following a moving subject that is easier or slower (person on a bike, jogger, large wading bird flying, pelicans or seagulls in flight, etc). I see no real difference in ability to track and pan with this type of action with either the EVF or the OVF...the action just isn't fast enough, small enough, or erratic enough to present difficulties. Sports on a field likely wouldn't be too difficult, unless using heavy telephoto where tracking with a fast moving person who is changing speed and direction often.
Once you get into very high speed motion, especially when introducing directional and speed changes, and a smaller target and a greater distance/longer lens, then panning/tracking with the target on an EVF can start to become more difficult than with an OVF. Note it's not tracking or panning while acquiring or not shooting that's the issue - that's not really any different, and there's no lag that should prevent following any target in an EVF. The problem comes when firing off a burst of shots. With an OVF, you get the effect of the mirror flipping up and down quickly, in between actual, real-time views of the subject - the effect is akin to blinking your eye - you don't lose the subject even when erratic...as long as your panning skills are up to snuff, you can stay with almost anything. With an EVF, when you fire the shutter, the blackout and restored electronic view is more interruptive than the blink of the mirror, and what's more, every EVF camera I've used or seen has what's commonly called 'slideshow effect' - where the frame just taken is displayed as the next frame is being shot - this happens in most of the burst speeds from 3fps to 7fps or so, and the effect can make it more difficult to pan and track with an erratic target, compared to an OVF. Again, if the subject is moving in a relatively predictable direction and speed, it's easy enough with the EVF to stay with the target just by using a steady panning technique. But if you're viewing that slightly staggered view of the last frame taken and trying to stick with a target that is slowing, speeding up, moving up or down, left or right, then that slideshow stagger can cause the target to start skewing out of the frame - sometimes you lose it completely, or sometimes you follow the view in the EVF and realize the subject has moved to the far left of the frame, so you overcompensate but by then the target altered again and suddenly it's half cut off on the right of the frame, so you try to compensate, and so on...because you're never seeing a real-time display of where the subject is, you're always guessing the next alteration to speed and direction based on the last frame you took.
What most EVF shooters do is learn to work their settings, and their panning technique, around this phenomenon. Once you've adjusted to it, you can generally use an EVF nearly as successfully as an OVF - again assuming OTHER issues aren't coming into play like lack of a CDAF focus system to keep up in continuous focus mode with your moving subject. There are still going to be a few areas where an OVF makes the job easier, or helps you get a higher hit rate...but it doesn't mean things are impossible with an EVF - it just takes a little more work, a few extra shots, and a lower hit rate. Take something very difficult to photograph, like sparrows...I've shot sparrows in flight with both EVF and OVF - so it's possible...but I definitely have a higher hit rate with OVF, and find I can track them a little more reliably and easily through an OVF, when firing off a continuous burst.
Hope that helps describe the difference a little bit for you!