i got curious and tried to look up these on Google but there is surprisingly little on anything other than chariot races.
Book 23 of the Iliad, believed to date from about 800 BCE, describes the funeral games held for the dead warrior Patroclus, the companion of Achilles. The events include both a chariot race and a footrace, along with archery, boxing, wrestling and spear throwing.
While not a historical account, it shows that both types of racing were practiced in the bronze age.