Incomplete answer; too long for a comment;
The speed and movemend can be considered in two ways: in one second, the ball "teleports" to a neighbor square (in one of the three directions), or the ball is considered to move like in the real life, continuously, with constant speed. Both problems are interesting, and maybe the first one is easy
Suppose the balls move continuously
My first thought is that if there is a ball moving horizontally and one diagonally, then they must meet. First, we can find the "times" the diagonal ball is in the strip containing the horizontal ball. Since these moments of time would be some irrational multiple of the period of the horizontal ball (the ratio is a multiple of $\sqrt{2}$), by a density argument, the balls must meet.
As in the previous answer, if all the balls have horizontal or vertical direction, the situation is easy. Suppose now that all the balls move diagonally, and each one moves in a rectangular strip with cut corners if the dimensions are equal, but when the dimensions are not equal, the trajectory can cover a big portion of the board. Each two such regions intersect, as can easily be seen, visualising the problem. The problem is the following: can we position the balls such that even if the trajectories meet, they are never simultaneously in the intersection? Even in a $8\times 8$ chessboard this problem seems hard, although the trajectories have the same length (when not moving on the great diagonal.
I will not continue with this case, since maybe this was not intended by the OP.
In the case where balls move "discretely" from a square to another, again, I feel that if the balls move diagonally, fewer balls can be positioned on the board.
[edit:] Ok, here's what came to my mind. I'll take for example the $8\times 8$ board. And take a rectangular diagonal loop and fill it with $14$ balls all moving in a "snake" style. We can make another loop on the other color and get another $14$ balls. I think this is an example $28$ balls. Don't know if the previous comments refer to this kind of structure, at least I didn't understand that.