I'm assuming when two sets of cavalry met they would both charge at each other, and the guys who got dismounted would end up fighting dismounted enemy troops, kind of similar to how the Battle of Bastards started in GoT.
EDIT: I'm thinking of cavalry engagements during the Punic Wars if it matters.
Do you have an era and region in mind? While there are some realities of cavalry combat that are universal, cavalry engagements between the Persians and Scythians were likely very different than cavalry engagements in the Polish-Soviet War.