I don't mind testing their app and helping improve the Hungarian. I use this app to practice other languages and those are very good, but they are good because they crowd sourced the right, most natural and most commonly used constructions. I'm just happy the added Hungarian!
"Beside the river" and to move "in into" are again sound very forced, unnatural and seems like someone tried to translate the Hungarian sentence word by word but not too successfully. "By the river" or "at the river" look better to me. "Move in" would do.