If you are not prepared, the change in gravity might cause you to fall, just like when you don't watch where you're going and accidentally step onto an escalator or other moving patch of ground. Heavy -> light might cause you to accidentally propel yourself forward too much; light -> heavy might cause your legs to give out on you because your muscles aren't prepared for the suddenly greater strain.
Based on the assumption that going through a rollercoaster doesn't cause any immediate health concerns through the changing gravity (unless you have a weak heart etc.) I assume that changing gravity within the space of a step will not cause any health problems either. Small gravity changes (ca. 0.1g) can also be compared to an elevator starting / stopping. So you can probably expect a sinking / lifting feeling when crossing the boundary.
Until your body and brain have adjusted to the new gravity, you will misjudge what strength it takes to run / jump / throw / lift unless you concentrate on those actions.
Normal-gravity comparison: When you lift a gallon bottle, your mind and your muscles expect it to have a certain weight. However, when the bottle is empty when you expected it to be full, you might accidentally knock it on the shelf above it before your muscles compensate for the unexpectedly light weight. On the other hand, you won't be able to lift it at first when you expected it to be empty but it turns out to be full after all.
Sure, your body will compensate quickly for unexpected weights. But in a chase situation where everything depends on split-second changes I expect you will do a lot of overshooting, undershooting, etc. The body works a lot with muscle memory, and until the muscle memory has gotten the memo 'different strength needed' you will have problems when you don't concentrate.
I am not a biology expert, but I'd say it takes at least 4-5 repetitions to set the muscle memory to the new standard. So, after 4-5 steps you should have walking/running down pat in the new gravity. But when you suddenly decide to throw in a pacours wall-jump you'll probably have to do a lot of on-the-fly correcting and landing in unexpected positions. After 4-5 pacours wall-jumps, you'll probably have that one down as well. But then you might decide to do some other acrobatics. The adjustment period will get quicker since you are constantly gathering experience (2-3 tries instead of 4-5). But it will take time and practice to completely get used to the new gravity in that regard (I'd expect somewhere between 30min and several hours).
Since you are writing a chase scene, the longterm issues are pretty moot. Medium long-term: just like some people can't adjust to living in high altitudes, there might be people who can't adjust to living in a different gravity. Really long-term: I'm thinking of growing up in low-G and then going to a high-G environment. Gravity affects things like bone density and muscle tonus (not only of the trainable muscles like biceps but also of intestinal muscles etc.), so I'd expect a lot of potential problems coming that way. (Just imagine how many problems Astronauts have coming back to earth after a stint on the ISS, and that is despite being there for less than a year and having a special training schedule to not lose their earth-G muscles).
Multiple gravity changes in short time issues:
Going with the rollercoaster example as quickly changing gravity, I imagine you might get a headache or an upset stomach from multiple rapid changes. If you have any pre-existing medical conditions, they might be exacerbated.
That aside, the longer the chase lasts, the more likely it becomes that one of the characters loses concentration and misjudges something in varyingly fatal or at least chase-ending ways. The quicker you change gravity, the less you can benefit from adjusting muscle memory, and the more you need to concentrate on every single step or motion.
I think it is possible to train for that, so that you know what to watch for when entering a new gravity at a full run. So someone trained might have an advantage over someone untrained. But unless gravity-caused medical issues come into play, it will come down to who does multi-tasking better (concentrating on running in new gravity + (chasing perpetrator or escaping chaser)).
When you are changing the direction of gravity (i.e. floor and ceiling suddenly becoming walls, and the walls becoming the new floor) you will probably have to deal with a lot of disorientation. And nausea.