What change of variables is this? I have come across this integral:
$$\int \frac{f(x+tv)-f(x)}{t}g(x)dx = \int -\frac{g(x)-g(x-tv)}{t}f(x)dx$$
which the author claims is justified by a change of variables, but I cannot see what they did. Would anyone be able to elaborate?

This is only correct in the limit as $t \to 0$ for definite integrals, and that's how it is used in the reference you are giving.
For general indefinite integrals and without the limit it's false. Consider for example the case $f(x) = x, \, g(x) = x^2$. The left hand side does not depend on $t$ in that case, but the right hand side is linear in $t$.
