Chase scenarios feature a Round Limit. At the end of every round, after every entity's turn is complete, increment a counter.
If the Round Limit is reached (by the counter value being equivalent to the Round Limit), then the scenario ends in failure. This could result in an adversary getting away, the PCs being caught, or rarely even death as a structure collapses atop the PCs. If the PCs are giving chase, typically only one PC needs to reach the Goal for success. However, if the PCs are being chased, all PCs must reach the Goal within the time limit to succeed. Collapsing structures and similar circumstances can result in partial success as the PCs who make it out in time are unscathed, where Injury or worse can await PCs who do not.
Some chase scenarios have Round Limits on each region. In these scenarios, if the PCs are fleeing, they lose if any PC is in a region when its round limit is reached. For life-or-death scenarios (such as collapsing structures), this results in the death of any PC in a region when its Round Limit is reached; any remaining PCs continue the scenario. If the PCs are giving chase, only one PC needs to make it through all the Round Limits to catch up with the quarry.
For more information, see Player's Guide Chapter 5: Exploration.