Differences in prognosis in HPV-positive and HPV-negative oral (oropharyngeal and oral cavity) squamous cell carcinomas (OSCCs) and increasing incidence of HPV-related cancers have spurred interest in demographic and temporal trends in OSCC incidence. We leverage multistage clonal expansion (MSCE) models coupled with age-period-cohort (APC) epidemiological models to analyze OSCC data in the SEER cancer registry (1973-2012). MSCE models are based on the initiation-promotion-malignant conversion paradigm in carcinogenesis and allow for interpretation of trends in terms of biological mechanisms. APC models seek to differentiate between the temporal effects of age, period, and birth cohort on cancer risk. Previous studies have looked at the effect of period and cohort on tumor initiation, and we extend this to compare model fits of period and cohort effects on each of tumor initiation, promotion, and malignant conversion rates. HPV-related, HPV-unrelated except oral tongue, and HPV-unrelated oral tongue sites are best described by placing period and cohort effects on the initiation rate. HPV-related and non-oral-tongue HPV-unrelated cancers have similar promotion rates, suggesting similar tumorigenesis dynamics once initiated. Estimates of promotion rates at oral tongue sites are lower, corresponding to a longer sojourn time; this finding is consistent with the hypothesis of an etiology distinct from HPV or alcohol and tobacco use. Finally, for the three subsite groups, men have higher initiation rates than women of the same race, and black people have higher promotion than white people of the same sex. These differences explain part of the racial and sex differences in OSCC incidence.