To assert that a trigger with an asynchronous process has successfully run, a developer should create all test data in the test class, use System.runAs() to invoke the trigger, then perform assertions. System.runAs() allows the developer to specify the user context to execute a specific block of code. This is useful for testing asynchronous processes that run in a different user context than the trigger, such as future methods, queueable apex, or batch apex. The developer can use System.runAs() to switch to the user context that runs the asynchronous process and then perform assertions on the expected results. Reference: [System.runAs Method], [Testing Asynchronous Apex], [Testing the User Context]