  Let $X$ be a normal compact K\"ahler space of dimension $n$. A surjective
endomorphism $f$ of such $X$ is int-amplified if $f^*\xi-\xi=\eta$ for some
K\"ahler classes $\xi$ and $\eta$. First, we show that this definition
generalizes the notion in the projective setting. Second, we prove that for the
cases of $X$ being smooth, a surface or a threefold with mild singularities, if
$X$ admits an int-amplified endomorphism with pseudo-effective canonical
divisor, then it is a $Q$-torus. Finally, we consider a normal compact K\"ahler
threefold $Y$ with only terminal singularities and show that, replacing $f$ by
a positive power, we can run the minimal model program (MMP) $f$-equivariantly
for such $Y$ and reach either a $Q$-torus or a Fano (projective) variety of
Picard number one.
