A 3-axis gimbal can keep your shot level and smooth no matter if you rotate the camera left or right, tilt it up or down, or rotate the camera on its roll axis. But it can’t help if the entire contraption moves up and down.
So when you walk or run with a gimbal, you move the camera and gimbal rig up and down naturally. And it's likely you'll see that up and down motion in all of your footage.
To minimize that up and down motion, shoot from your chest, keep the gimbal close to you, lock your elbows and wrists, and walk with light steps. Essentially, walk like you’re trying to sneak into your house without waking anybody up. Take short steps, rotate on the balls of your feet and heel rather than stomping down with your flat feet, bend your knees, and try to stay relaxed. Don’t worry about what you look like when you’re shooting.
If you have a gimbal that has a one-handed mode, or is able to rotate into a sideways or "briefcase" mode, you can also try shooting with the gimbal very low to the ground. You naturally walk or run more carefully as you try to keep the camera rig from hitting the ground.
If you are in two handed mode, another thing that helps to minimize the up and down motion is if your rig is heavier in general. The more weight, the less it’ll move up and down, just by its nature. So if you want to create running shots and have a choice of camera, lens, or gimbal you can use, go for the heavier gear.
Finally, one surefire way to eliminate that up and down motion is to simply not shoot while walking or running. Get into a car, or on a bike, or in any number of moving platforms. The old wheelchair trick still applies, or getting pushed on a skateboard, or even getting onto an electronic skateboard.