I am looking at the paper in the title and I am trying to derive their result in equation $2$. Here is what I obtain. Start with:
$$X(ct) \stackrel{d}{=} M(c)X(t)$$
where $M(.)$ and $X(.)$ are random independent functions of parameters $c$ and $t$. Set $f(t) = ct$, then $f(c^{-1}t)=t$, then
$$X(f(t)) \stackrel{d}{=}M(c)X(f(c^{-1}t))$$, moving along the time axis such that the second term is $t+\Delta t$, we have
$$X(t+f(\Delta t)) \stackrel{d}{=}M(c)X(t+f(c^{-1}\Delta t))$$
or
$$X(t+c\Delta t) \stackrel{d}{=} M(c) X(t+\Delta t)$$
Now subtracting $X(t)$ from both sides:
$$X(t+c\Delta t) - X(t) \stackrel{d}{=} M(c) X(t+\Delta t) - X(t)$$
But the result in the paper is:
$$X(t+c\Delta t) - X(t) \stackrel{d}{=} M(c)(X(t+\Delta t) - X(t))$$
