There's no sense of moving TO a location beside the river, in this translation. You could just be beside the river, moving (in whatever manner). But of course it doesn't make sense to say "moving to beside the river." You'd need to introduce more words that aren't in the Hungarian sentence, but maybe are implied?
It sound weird to me. If I say that I'm going to, moving to, running to (etc, whatever the verb is) a place, then in my mind, that place has to be a name of some sort. Either a the name of city, country, street, apartment complex, etc, or a type of place (a store, a school, a river...). But to say that I'm moving "to beside" something sounds - if not wrong, then not very natural. "Beside" isn't a place that you can go to.
I could say that I'm moving to a place beside the river. That's not in the original sentence, but it's how I would translate it if I thought it would be accepted here.
As suggested by Bastette54, “Do you move to (a place) beside the river,” “a place” is implied. That said, as another native English speaker, I agree that “move to beside” makes more sense than “move beside” and it gives voice to the meaning of költözni. Though in English I’d more naturally say “Are you moving to (a place) by the river,” and most naturally, and with no small measure of jealousy, “Are you moving next to the river?”
Again, agreed. "Do you " implies habitual actions, eg, "Do you go out dancing on Saturday nights?" But if you want to know if someone is out dancing right now, you ask, "Is she out dancing?" Similarly, "Are you moving to..." means "Are you in the process of moving?"It can also mean "Are you about to move?" (in the near future).
One would never ask, "Do you move to the country?" about a current or upcoming move. It would only be a valid question if you wanted to know about a habitual action, such as, "Do you move frequently?" (Say, for your job, military, etc.)
"Mellett" is moving along the river. It stays next to the river. Sooo... if you are moving, but both your old and new places are beside the river, it is possible.
And "mellől" is "from beside". "Away from a location next to". That is, you had enough of the river view, you are moving to the desert.
There are differences between some of these expressions, and some are synonymous. I'll throw in the preposition "at" here, as well as add an example of "building" to make a bit better difference.
at the building - You're very close to the building and have things to do there. It might be a meeting point ("I'm at the building, where are you?") and it's likely that you plan to go in. In case of the river, you might have planned to go for a swim. This is translated with -nál/-nél in Hungarian - a folyónál, az épületnél.
by the building - You're very close to the building, but you don't have plans to interact with it. You might "wait at the corner by the building" or "walk by the river". This is also translated with -nál/-nél in Hungarian.
next to the building/beside the building - This is a very defined location. You're at the side of the building - left or right. For instance "The theatre is next to the building" means that it's the building's neighbour. They share a wall. For rivers it doesn't really make a difference if you use "by" or "next to", because in natural circumstances you can only be at the side of a river. "Next to" and "beside" are translated with the postposition mellett - a folyó mellett, az épület mellett.
close to the building/near the building - You're in the general area of the building, but generally further away than "by". For instance, I live close to the zoo in my city, but I can't even see it from my street, so I wouldn't use "by". It's just a general direction/area. If you are "close to" a river, you might still have to walk or drive some distance to get your feet in the water. "Close to" can also be used if you're moving towards a goal: "I'm close to the city now." "Near" is sometimes even more fuzzy than "close to", but they're generally used synonymously. These are expressed with "közel vmihez" in Hungarian - közel a folyóhoz, közel az épülethez.
If you stick to the pairings of
- at, by - -nál/-nél
- next to, besides - mellett (and friends)
- close to - közel
you'll be fine throughout this course. :)