fourball order of play

Question:

I know that a fourball team may play in any order they decide but do both play regarless of position. Example if A and B are playing against C and D. All four hit on the green. A is furthest from the hole. Be is closest to the hole. and C and D are in the middle somewhere. A is away and has the right to putt first...now does B put second or does C and D get to putt in order.

If A putts first, and the ball finishes closer to the hole, than either of C or D, then it is the turn of C or D.

If A or B continue to putt out then they are deemed to have played 'out of turn' and C/D can demand they replace their ball, and wait until their correct order of play.

i guess i am misunderstanding the rule then...because i thought the TEAM played in order with the order of play within the team up to them.

If A was away I thought he would putt first...then B as the other member of the team would putt. Then C and D Would do likewise. I understood that if in fact B did not put after A that he was picking up his ball...and conceding the hole unless he gave up his right to putt before C and D and they agreed.

I know that if the team wins the hole BOTH partners Tee off first on the next hole. But when through the green it is simply the person furtherest from the hole who hits first. Kind of confusing for this simple mind.

thanks
yo