  The one-dimensional polynuclear growth model with external sources at edges
is studied. The height fluctuation at the origin is known to be given by either
the Gaussian, the GUE Tracy-Widom distribution, or certain distributions called
GOE$^2$ and $F_0$, depending on the strength of the sources. We generalize
these results and show that the scaling limit of the multi-point equal time
height fluctuations of the model are described by the Fredholm determinant, of
which the limiting kernel is explicitly obtained. In particular we obtain two
new kernels, describing transitions between the above one-point distributions.
One expresses the transition from the GOE$^2$ to the GUE Tracy-Widom
distribution or to the Gaussian; the other the transition from $F_0$ to the
Gaussian. The results specialized to the fluctuation at the origin are shown to
be equivalent to the previously obtained ones via the Riemann-Hilbert method.
