Continuous Estimated Time of Arrival
If I understand correctly (not sure), it looks to me like you don't need a model which can predict at any time, you just need:


a model which predicts the ETA at any stage given information about the past stages. The easiest way to do that is probably to just train a different model for each stage, since the number of stages is fixed.
Then between two stages the ETA can be updated in a deterministic way: if the last stage was passed at time $t$ and the predicted ETA was say 10mn, then at time $t'$ the ETA is just 10mn - $(t'-t)$.