Is the decimation process time invariant?
Sample rate conversion systems (expansion or decimation) are time-varying operators.

For your example system of decimation by $M$ :

$$ y[n] = T\{x[n]\} = x[Mn] $$

you can easily see that results of shift in the input and output are not the same; i.e.,

$$ y_1[n] = T\{ x[n-d] \} = x[Mn - d] $$

and 

$$ y_d[n] = y[n-d] = x[M(n - d)] \neq y_1[n]$$