Florence to Pisa airport after our Heart of Italy tour ends in Florence?

Train to Pisa Centrale, then the people-mover shuttle train to the airport.
http://www.pisa-airport.com/en/the-passengers/transport/people-mover.html

Virtually all trains going from Florence to Pisa Centrale are regional trains, with the RV (regional veloce) trains being slightly faster. Most trains are direct, but a few require a change in Empoli. It's easy to buy regional train tickets at the ticket machines in the station, and they're good for any regional (including RV) train on the date indicated on the ticket. The ticket machines do give change, so you could use a 20 euro bill to pay the 16.80 for two tickets and would get 3.20 in change. Important: make sure you validate the tickets before boarding. If you have any early departure to Pisa and want to save a few minutes at the station in Florence, you can buy your tickets a day or two in advance. Just be sure to change the date to your actual day of travel as it defaults to current date. Last time I used a ticket machine to buy regional tickets in Florence, it prompted me to confirm the date.

Besides the train+people mover, you may want to consider the AUTOSTRADALE bus from Florence (in front of the train station) to Pisa airport (in front of the terminal).

All my Florence friends prefer the Autostradale bus. It’s 1h10min journey (therefore in most cases shorter than the train+people mover) but with no need to transfer modes of transportation while carrying your luggage. You give the luggage to the driver who puts in the belly of the bus when you depart, and you retrieve it when you get off the bus in front of the airport terminal. It’s as easy as with a private car or airport shuttle service.

Autostradale Bus timetable below
https://autostradale.it/Images/Linee/Orari/142.pdf

Bus booking below
https://autostradale.it/Ricerca.aspx

Besides the train+people mover, you may want to consider the
AUTOSTRADALE bus from Florence (in front of the train station) to Pisa
airport (in front of the terminal).

From one of my previous posts:
Unless they've settled the issue, the city is trying to divert the buses from the stop right in front of the airport to the Pisa Mover station. See my comment at the bottom of this thread:

https://community.ricksteves.com/travel-forum/italy/pisa-florence-help

Doesn't mean you shouldn't take the bus ( we have; it's very comfortable) but to be prepared to take the P.M. from station to airport if that's where they drop you off.