Long time solutions to the Kähler-Ricci flow occur in the final step of the analytic minimal model program proposed by Song and Tian. In this talk, we will describe the behaviour of curvature in this setting. In particular, we show that the singularity type of long-time solutions are independent of the initial metric. We will also talk about some extensions of this result to the non-Kähler case, that is, the Chern-Ricci flow.