  Starting from $\mathbb{C}^*$-actions on complex projective varieties, we
construct and investigate birational maps among the corresponding extremal
fixed point components. We study the case in which such birational maps are
locally described by toric flips, either of Atiyah type or so called
non-equalized. We relate this notion of toric flip with the property of the
action being non-equalized. Moreover, we find explicit examples of rational
homogeneous varieties admitting a $\mathbb{C}^*$-action whose weighted blow-up
at the extremal fixed point components gives a birational map among two
projective varieties that is locally a toric non-equalized flip.
