There was a story that 2 cohanim ran up the ramp and one pushed the other and breaking his foot. Since they saw that it was becoming dangerous, the chachamim stopped the system of running, and only did the lottery.

There were 4 lotteries made each day and terumas hadeshen was the first.

Yuma 2:3

The second lottery was for the slaughtering of the korban tamid, to determine who would sprinkle the blood, who would clean the ashes from the inside mizbeiach, who would clean the menora, and who would bring the limbs to the ramps. The limbs were brought in the following order: the head, the back feet, the two front feet, the tail, the left hind leg, the breast, the neck, the 2 sides, the intestines, the fine flour, the chavitim, and the wine. All together it took 13 cohanim. Ben Azai said the limbs were brought up in the way the animal walks.