If the players are

A - Ineffable

B - ChrisCMU

C - malakkan

D - skandalo

Then the first three turns go

A, D, C, B

B, A, D, C

C, B, A, D

This all looks like the correct cycle. First ADCB, then the B-ADC (the B moved from the right to the left.) Then CBAD (the CB moved from the right to the left.)

For example: ABCD becomes BCDA for the next turn, followed by CDAB, and finally DABC before starting back at ABCD.

Turn 9 blue still moved before me, so he moved ahead of me 3 turns in a row. I don't see how this is possible.

Focus on two letters. Let's say, A and B. Given the way it cycles, A comes before B in 3 out of the 4 options (ABCD, CDAB, DABC) and comes after B in one (BCDA). So if A and B are fighting, A will move first 3 times in a row, then after B once. I agree it's not very fair, and I don't recommend using cyclic move order in team games for this reason.

Another way cyclic move order could have been implemented is to reverse the cycle each turn. For example, swap between ABCD and DCBA. This would ensure that if two players are fighting, they'll alternate first moves each turn. This has its own drawbacks, however, namely that the same players will always get first or last move, and other players will always get middle-moves. If this is something you want, you should create an entry for it on uservoice. I'm not sure if there's a better option than either of these - namely a predictable way to cycle that doesn't have any drawbacks.