They became either armoured units or mechanised infantry.

See the Blues and Royals, a cavalry unit that was converted to armoured recon in the modern theatre. They also serve as ceremonial troops.

It's a big deal disbanding historic military formations. When the NZ airforce terminated it's Air Force strike capacity the colours had to be put up with lots of pomp and ceremony.

There are multiple historic American cav formations that have new roles- first cav became first air cav. 11th black horse became the 11th armoured cav regiment, which famously served as the tripwire on the fulda gap during the worst part of the Cold War.