How to solve this focusing issue on my DSLR?

I filmed the contents of the following video with a Nikkor 35 mm prime lens using Nikon D3200. I set the focusing mode to Auto. And the sights in the video were constantly alternating between in-focus and out-of-focus.

I wonder how I could I have brought everything in focus all through video without it going in-focus then out-of-focus then in-focus?

(Manual focusing would not have solved the issue, for in that case I would have to manually adjust the focus every few seconds)

You answered your own question. You have a scene that is cluttered with people. I'm amazed that the camera did that well on auto focus. Use manual focus and use a smaller aperture so your depth of field is wide enough where you don't need to focus as much.

Use manual focus.

Your camera was doing exactly what is supposed to do -- trying to determine exactly what part of your constantly moving scenes to focus on. If you want to reduce this effect, you should use a larger f-stop (smaller aperture) so that you have a larger depth-of-field (DoF) and a larger part of your scenes in focus.

It has always been better to manually focus during video. There are two ways to focus manually, with your hand on the focus ring of the lens and with your feet and legs to walk.