In my experience for the case of tuning of neural networks during training with an early stopping mechanism, the latest reported metric before early stopping is used.
The metrics reported in the meantime are used for the eventual schedulers like ASHA for choosing the eventual interruption of the training if it performs below the population median.
Each call to tune.run() will produce a result, and each result is processed by the search algorithm, the scheduler, and callbacks (e.g. loggers). Schedulers can at any point in time decide to stop a trial (e.g. for early stopping like in ASHA), otherwise it will just continue training.
Generally after each result a decision about the fate of the trial we be made. Most of the time this is to just continue running.