And here are the races that I ran:

Race 1:
Spoiler Alert, click show to read: 
No cheating

Out of the Gate Order:
7 Blue
5 White
4 Red
2 Green

Lap 1:
Green overtakes Red

Blue
White
Green
Red

Lap 2:
White overtakes Blue

White
Blue
Green
Red

Lap 3:
Blue overtakes White – COLLISION – Blue Chariot damaged, overtaking aborted, future O rolls take +1

White
Blue
Green
Red

Lap 4:
Blue overtakes White
Red overtakes Green

Blue
White
Red
Green

Lap 5:
Red overtakes White (due to White tired horses)
Green overtakes White (due to White tired horses)

Blue
Red
Green
White

Lap 6:
Blue
Red
Green
White

Lap 7:
Green overtakes Red

Blue
Green
Red
White


Race 2:
Spoiler Alert, click show to read: 
1095 AD – 4 Horse race

No Cheating (Both teams have Honest charioteers)

Out of the Gate Order:

Blue 5
Red 5
Green 2
White 5

On reroll

Blue 6
Red 5
White 5
Green


On second reroll

Blue
Red 1
White 5
Green

Final Order:
Blue
White
Red
Green


Lap 1:
White overtakes Blue – No collision roll needed
Red overtakes Blue – No collision roll needed

White
Red
Blue
Green


Lap 2:
Red overtakes White

Red
White
Blue
Green


Lap 3:
Green overtakes Blue

Red
White
Green
Blue


Lap 4:
Blue overtakes Green

Red
White
Blue
Green


Lap 5:
White overtakes Red – No collision roll needed

White
Red
Blue
Green


Lap 6:
Blue overtakes Red

White
Blue
Red
Green


Lap 7:
Red overtakes Blue
Green overtakes White – Unlucky reroll results in failure to overtake

White
Red
Blue
Green


Race 3:
Spoiler Alert, click show to read: 
1132 AD – 4 Horse race

Green Faction, Green Team cheats (via pevergreen) with additional cheating of Green Team halting before finish line if he is in first place. – Whips all overtaking

Out of the Gate Order:

5 Blue
4 Green
3 White
2 Red

Lap 1:
Green overtakes Blue in dominating fashion (huge bonus)

Green
Blue
White
Red

Lap 2:
Blue overtakes Green, Green whips Blue but it does not stop overtaking.
COLLISION – Blue chariot damaged and horse injured. Overtaking is aborted and -1 to all future O rolls for Blue.

Green
Blue
White
Red

Lap 3:
White overtakes Blue
Red overtakes Blue

Green
White
Red
Blue

Lap 4:
No Change

Green
White
Red
Blue

Lap 5:
White overtakes Green, but fails by bad luck.

Green
White
Red
Blue

Lap 6:
Red overtakes White

Green
Red
White
Blue

Lap 7:
Red overtakes Green, Green lets Red go past him without much struggle.
Blue overtakes White

Red
Green
Blue
White