Pawns on infinite chessboard

Let $k$ be a fixed integer. The cells in a $1 \times \infty$ chessboard is colored in the following fashion: ..., red, blue, green, red, blue, green, ... (and so on) There are a number of pawns on the board. At each turn, we are allowed to choose two pawns $A,B$ and make $A$ "jump over" $B$ so that the new distance is $k$ times the old distance. Formally, we may choose $A$ and $B$ with distance $d$ and move $A$ so that the new distance is $kd$ and $A$ is on a different side of $B$. Multiple pawns may occupy a single cell. Determine all values of $k$ such that, regardless of initial configurations, it's always possible to move the pawns to all occupy the same color.