Long-awaited story boards!

[EDIT: Ah, crap, the images are far too wide for this skinny column.  Click them to see the full image, so that the explanations make sense.]

The aim of these diagrams - which are not necessarily intended to represent proposed visual design - is to show the progression of a sample game at various points.  So here we go:

Initialization
Set up requires choosing a provided closed-loop racetrack (of relatively consistent width) on grid paper, or drawing one yourself.  A start/finish line and direction of travel (see the arrow beside the start line) are decided upon, and each player chooses a symbol and selects a position on the start line.
The possible first moves are shown (for the blue-circle player) as filled blue circles.  Since clearly they have not started moving - a speed of (0,0) - their next move can have a speed of -1, 0, or 1 along each axis.  All possible moves are shown here for completeness, but obviously, you would never move backwards behind the starting line.

Getting going: how to move
At this point, each player has taken 3 moves, and the options for the fourth move are shown for each.  (Dual-color circles indicate that that point is an option for both of those players.)  The basic rule is that for any move, a player can hold constant or change by 1 the number of points to move along each (horizontal and vertical) axis.
For example, if in the prior turn, a player moved (3,2) in (horizontal, vertical) movement, their next move could be any of these options (within bounds): (2,1) (2,2) (2,3) (3,1) (3,2) (3,3) (4,1) (4,2) (4,3)
Note a couple things:

• Players cannot occupy the same grid point at the exact same time.  (BUT, any player can occupy a point that was occupied in a past move by another player.)  Taking a grid point that would be ideal for another player's move could be a strategy.
• Players are not allowed to move to grid points outside the track (obviously).  Clearly, here, the green and blue players have a decreased set of options.  Grid points exactly on the border of the track are considered in-bounds.

Stuck
Now each player has taken 6 moves.  The red player did not think ahead enough and has created a situation in which all possible next moves are located out of bounds.  At this point I have not decided how to handle this situation, but I do not think cutting the player out of the game from then on is ideal.  That's no fun.  I may decide to require some sort of backtracking to a prior position.

The end: finishing/winning
Here, the completion of the game is shown (except the blue-circle player, left out to make the example easier).  Assuming that the red and green players were taking the same turn at the bottom-right of this quadrant of the diagram, notice that 14 moves for each are shown here.  Since the red player reached the start/finish line (and the green player has not yet), the red player wins.
Notice this diagram includes an example of players occupying the same grid point at different times in the game: the green player's 10th move on this diagram is the red player's 8th move.