Just a couple of things to note:1. It's not "Bonus" Action it's "Bonus Action." The difference is each turn you can perform 1 Move Action, 1 Action, 1 Bonus Action and 1 Reaction.
The Reaction is granted when your character reacts to things, for example Attacks of Opportunity. This is where the old "1 AoO a turn" is hidden, because you only get 1 Reaction until it's your go again.
The Move Action, as you rightly believed, can be split between Attacks of an Attack Action.
The Bonus Action is where a lot of people trip up, because they see Bonus and don't read it as a label (It could be called Sasquatch Typhoon and still have the same effect). Your fellow Monk gets to spend 1 Ki to flurry of blows after taking an Attack Action, this is when it technically gets tricky.
1) As part of the attack action, you can move fight move. So you could Move 10', attack, move 5', extra attack, move 20'.
2) When the attack action is over you then get to flurry of blows. But the tricky part is it says immediately after you take the attack action. 5th edition is full of rules like this which technically don't have definite timing - in this instance when do you "take" the attack action, is it from your first hit, or when you declare it?
Personally I'd let your friend do this:
1) After his first attack he can spend Ki to flurry of blows, and define "taking the attack action" as being when you make your first attack roll.
2) After flurry of blows he may now continue the rest of his action and move, make his extra attack and move some more.
3) Remember you only get 1 Bonus Action a turn and cannot Martial Arts + Flurry of Blows in the same turn. If you use Martial Arts there is no timing involved with the Bonus Action, so you could easily allow your friend to Move, Attack, Move, Extra Attack, Move, Bonus Attack with Martial Arts against 3 separate foes (the lack of immediately would easily allow this, even if some people interpret the rules differently).
So ultimately the Monk can achieve 2 Attacks and 1 Flurry of Blows for a total of 4 attacks in a round, 3 of which occur at the same time (and you are free to rule they must all attack the same foe too!)
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