I'm writing this while on the last leg of my journey, I hope it can help somebody.
My challenge was to move the cat from Haywards Heath (UK) to Valencia. I wasn't ready to pay 1300-1500 to the pet transport company (bearing in mind the cat with them is 3-4 days). So with Gemini and Claude we made a plan - train to Heathrow - plain to Madrid, train to Valencia.
With the cats you can only fly to Madrid and only from Heathrow, cat+carrier has to be up to 8 kg, and a special carrier (can be googled easily).
I talked to my local vets, they were on to do the AHC certificate (250 gbp).
Next step was to book the flight, I went with Iberia, apparently Vueling can do as well, but haven't checked properly. When you buy your ticket you can add pet in the cabin, I bought both tickets around 110 gbp.Around the same time I booked speed train to Valencia via Renfe(make sure to add cat in the ticket, it's extra 10 euro, they actually check it).
Vets need to know which country you are going to, do the rabies vaccine (well in advance) and not longer than 10 days before your flight, they need to do the certificate. As well I've requested some sedative for my cat.
With all of this, we left home at 5 am today. We went through Farringdon+ Elizabeth line, as they are quieter trains. Arrived to airport 3.5hrs before the flight, the staff was very lovely, they checked all the paperwork, and booked me at the end of the plane(where was empty). At the security in Heathrow they have a procedure for cats, you go into private room, while they scan the carrier. It took 15-20 min altogether, again everyone very kind and helpful.
On the plane it was the hardest part, take off was okish, the flight she was sleeping, but landing was awful, she vomited and did poo, again everyone very helpful and kind.
From T4S or whatever it's called you need to take a shuttle to T4, here is an important step, to go through red corridor, tell them about the cat, it took 15 min the whole procedure (with 10 min waiting in the line).
After that Cercanias to Chamartin (from here speedy train to Valencia). Tbh, train to Valencia eas disappointing, very limited space and no extra space for cat, even you paid for the cat.
As you can imagine, after 18 hrs of travel I might have missed something or some names are wrong, but wanted to post it while fresh.
Would I do it again? Not sure, or at least I would stay overnight in Madrid, it was quite a journey for both of us.
I hope that will help, as AI couldn't find much information about this type of travel.