If I recall from the last time I played multiplayer, Each player that controls a character that has a reaction available and appropriate at a particular trigger gets a prompt; if one player controls multiple characters with appropriate available reactions for the trigger, then they are all displayed in the same box. Each player makes their choice, while, at the bottom of the screen, the turn notifications denote that the other player is considering a reaction, until all choices are input (in real time, in multiplayer games I've played, this takes less than a second to go through the vast majority of the time, and has no impact on the feeling of turn progression compared to other elements, such as simply watching other players or enemies take their turns), at which point they all play out.
Reactions triggered by movement take into account the creature's entire planned movement - so if it would trigger one when it starts to move, and another ten feet away, it doesn't pause the game twice - only once for both. If the second reaction proves unnecessary (because the creature is killed by the first, for example) the reaction isn't consumed.
