Unless you just handwave everything, there is no physical sense to viewing any spatial dimension "different" from any other, or to be able to travel "through" an additional dimension.
Flatland is a great example. If you take a Flatland style piece of paper (2D) and a second piece of paper, and put them on top of each other, you do not get 3D space. Topological, you are still locally 2D with some weird effects like being able to "tunnel through" between the sheets of paper by making holes and glueing the edges together, or forming a donut out of a single sheet by glueing opposite edges, or being able to flip beings by glueing a band of paper in the Moebius configuration.
You can do a little experiment: Take a sheet of paper (15x15cm in size) and draw a 10x10cm square on each with a pen. Make it one of those pens which "bleed through" the paper, and thin paper. So it seems like the drawing is really "inside" the paper, just like a Flatlander would be. Now stack many of them, until they build a 10cm high tower. You now have a 10cm high tower of 2D sheets, but what you do not, in no fashion whatsoever, have, is a 10x10x10cm 3D cube.
Thinking the other way round, no 2D Flatlander could ever "lift off" its universe to travel through 3D space, in the same sense that you could never lift off a pencil drawing from its sheet of paper. The operation would just not be defined in any useful mathematical or physical sense.
Our (real) 3D is fundamentally different from a hypothetical 2D stacked upon each other. Even if you take our greatest microscope, at no time does anything fundamentally behave in a 2D way.
A more mathematical way to see how this does not make sense is to see that all dimensions (to be called dimensions in any way making physical sense) would have to be continuous. The stacking 3rd dimension would be discrete and thus people would probably not even call it "space".
All of this scales up seamlessly to 4D.
What you absolutely could do would be to bend 3D space in creative ways. This is 100% compatible with our real world, as far as we know. This is what happens in the Flatland example, of course - stacking two sheets of 2D paper on top and glueing them is just an easier to imagine way of having a single 2D space in the first place and bending that one into a stacked configuration.
But you cannot have a (physically working) 4D space somehow "embedding" a 3D space and having meaningful travel for the 3D objects.
Your stacking version could also just be a multiverse as we are thinking about today. Those are perfectly happy (and basically required to be) stacking/discrete. But don't call them 4D please, either.
...you can also just forget everything above, you have artistic freedom. It would just grate hugely on everybody who has a passing intuition about these things, spoiling suspension of disbelief absolutely. The broad masses won't probably notice anything, I guess. ;)