Despite the appeal of screening travellers to prevent case importation during infectious disease outbreaks, evidence shows that symptom screening is largely ineffective in delaying the geographical spread of infection. Molecular tests offer high sensitivity and specificity and can detect infections earlier than symptom screening, suggesting potential for improved outcomes. However, they were used to screen travellers for COVID-19 with mixed success. To investigate molecular screening’s role in controlling COVID-19, and to quantify the effectiveness of screening for future pathogens of concern, we developed a probabilistic model that incorporates within-host viral kinetics. We then evaluated the potential effectiveness of screening travellers for influenza A, SARS-CoV-1, SARS-CoV-2, and Ebola virus. Even under highly optimistic assumptions, we found that the inability to detect recent infections always limits the effectiveness of traveller screening. We quantify this fundamental limit by proposing an estimator for the fraction of transmission that is preventable by screening. We also demonstrate that estimates of ascertainment overestimate reductions in transmission. These results highlight the essential role that quarantine and repeated testing play in infectious disease containment. Furthermore, our findings indicate that improving screening effectiveness requires the ability to detect infection much earlier than current state-of-the-art molecular tests.