We construct new families of K\"ahler-Ricci solitons on complex line bundles over $\mathbb{CP}^{n-1}$, $n\ge2$. Among these are examples whose initial or final condition is equal to a metric cone $\mathbb{C}^{n} / \mathbb{Z}_{k}$. We exhibit a noncompact Ricci flow that shrinks smoothly and self-similarly for $t<0$, becomes a cone at $t=0$, and then expands smoothly and self-similarly for $t>0$; this evolution is smooth in space-time except at a single point, at which there is a blowdown of a $\mathbb{CP}^{n-1}$. We also construct certain shrinking solitons with orbifold point singularities.