Imagine you’re walking down the street, and a flood comes. A big wave of water hits you, picks you up, and the next thing you know, you’re sitting on the sidewalk. You can say that the water swept you up when it washed/moved over you.

picks you up

Вот мне и непонятно, почему обязательно moves OVER your - как будто ты обязательно остаешься внизу, на дне, на месте

It moves over you because it’s higher than you, over your head when it hits you.

Otherwise, we would say that it crashed into you… if it was shoulder height or waist height.