I'm assuming you're talking about the Training Carriers (aka AVT's) used for Student Naval Aviators and normally stationed off JAX or more frequently off the Texas Gulf Coast where the advanced training bases were located.
In my day, AVT-16 Lexington was used. "Lex" got replaced by the Forestal and is now on display at Corpus Christi.
Forestal did her turn until the early 90's when decommissioned and later scrapped.
Since then, the Navy hasn't had a designated AVT, preferring to train Student pilots their carrier landing skills in simulators.
I suppose the occasional CVAN may slide down by JAX for old times sake but I'm afraid the times when the Lex or Forestal were there ended 30 years ago.
So if there's a CVAN (supposedly with the rest of the Battle Group) or something else requiring the services of a Greyhound out there, it ain't standard by any means.
Not speaking of visiting carriers, and technically not Navy, but Naval Station Mayport (US 4th Fleet HQ) is also the homeport for Marines USS Iwo Jima (LHD-7).
Most would look at that deck, and air traffic, and refer to the Iwo Jima as a carrier, while it is just a US Marines amphib assault ship. And it does spend a lot of time off Florida’s East Coast. It loiters out there, with regular flights from multiple Air Stations from mostly NAS Jacksonville, but really from all over the Southeast US.
However, while I have been working well North at MCQB, and I am aware of significant increased naval carrier group training exercises over the last several years off our Southeast Coast, and Florida. I know the Abraham Lincoln has spent considerable time out of Jacksonville over the last few years, with heavy f-35 training/joint training. Won’t go specific, but some of it hasn’t just been training either, either along our East Coast, or especially further South and West.
It’s not normally involving my lane from MCBQ, belvoir, oni, “langlang”, but JAX has become far more important over the past four years. Necessarily significantly heavier traffic between Norfolk and JAX than most know, even while JAX is smaller.