  We develop an isotopy principle for holomorphic motions. Our main result
concerns the extendability of a holomorphic motion of a finite subset $E$ of a
Riemann surface $Y$ parameterized by a point $t$ in a pointed hyperbolic
surface $(X, t_{0})$. If a holomorphic motion from $E$ to $E_{t}$ in $Y$ has a
guiding quasiconformal isotopy, then there is a holomorphic extension to any
new point $p$ in $Y-E$ that follows the guiding isotopy. The proof gives a
canonical way to replace a quasiconformal motion of the $(n+1)^{th}$ point by a
holomorphic motion while leaving unchanged the given holomorphic motion of the
first $n$ points. In particular, our main result gives a new proof of
Slodkowski's theorem which concerns the special case when the parameter space
is the open unit disk with base point $0$ and the dynamical space $Y$ is the
Riemann sphere.
