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I would like to apply a coordinate transformation, visualised as an action on a two-dimensional grid like

grid

The particular example above is $$x^\mu \mapsto \frac{x^\mu - b^\mu x^2}{1-b\cdot x+ b^2 x^2}.$$

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If you want a point-to-point transformation:

f[x_: List, b_: List] := (x - b (x.x))/(1 + (b.x) + (b.b) (x.x));

myinput = DeleteDuplicates@
   Flatten[Union[Table[{i, j}, {i, 0, 10, 1}, {j, 0, 10, .01}], 
     Table[{i, j}, {i, 0, 10, .011}, {j, 0, 10, 1}]], 1];

ListPlot[myinput]

enter image description here

ListPlot[f[#, {.5, .1}] & /@ myinput]

enter image description here

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