I am not sure if anyone has examined exactly what you are looking for, but there has been some work that has considered the difference in impacts between persisting (e.g., multi-year La Nina) and transitioning (e.g., El Nino "switch" to La Nina) ENSO states. There potentially are differences between the two scenarios because the ocean conditions outside the tropical Pacific tend to be distinct, and that can affect the impacts. 

The studies that come to mind are this one and this one. I am sure there is more work to be done, as there has been less work on ENSO teleconnections during the season when rapid transitions tend to occur.