These creatures would have the ability to move through the fourth (or higher) dimensions in multiple directions.
Explanation: When we think of moving through our 3-dimensional space, we think of the directions we can go, forward, back, left, right, up, down, barrel-roll, other rotations, etc.
If a creature were able to move through other dimensions in the way that we move through these three, then people-like-us (that is, creatures usually able to move through only three dimensions continuously, and a fourth (time) in only one direction) would experience those creatures as traveling in ways that would be difficult for us to really understand... possibly seeming to be instantaneous.
In Liu Cixin's Three Body Problem series, increasing dimensions was described as an "unfolding" of reality - like opening a folded piece of 2D paper into a 3D origami creation (and yes, real paper is 3D, this is metaphor), whereas decreasing dimensions was... uh, apparently either really depressing or extermination, if the creatures natively existed in more dimensions? No spoilers.
As far as catching a ride, this is where I start waving my hands, speculating... maybe these creatures eat food that is anchored in the 3 spatial dimensions, and their food travels through dimensions with them, so people figured out a sweet-tasting suit to let them be eaten without being consumed. This would allow them to travel inside the creatures and get, ahem, dropped off at the other side?