As most responses mentioned, it depends entirely on the airlines.
In my case, I will be flying with Qatar Airways (South America to Asia) with my 1 year old cat. They do not require any private companies to do the logistics. All they require is that the animal has the valid health certifications from the local authorities, fulfil the requirements of the regulations of the country you're flying to (ie the health certifications or if your pet requires a microchip, etc), have a big enough carrier (details on their website) and also pay the excess luggage (since pets - with exception of falcons and service dogs - are not allowed in the cabin). The Qatar Airways website also mentioned that if you are transiting in Doha, their ground staff will look after the pet (walk it, check for stress or sickness, make sure their food and water is refilled, etc).
When I talked to the Qatar office in Buenos Aires, they also require that I call them 3 days prior to the flight to reserve the space in the cargo of the plane. I know the travel time is over a day, but I am not worried since I know that someone else has travelled the same route with a cat a few years back and everything was fine.
In my case, I will be flying with Qatar Airways (South America to Asia) with my 1 year old cat. They do not require any private companies to do the logistics. All they require is that the animal has the valid health certifications from the local authorities, fulfil the requirements of the regulations of the country you're flying to (ie the health certifications or if your pet requires a microchip, etc), have a big enough carrier (details on their website) and also pay the excess luggage (since pets - with exception of falcons and service dogs - are not allowed in the cabin). The Qatar Airways website also mentioned that if you are transiting in Doha, their ground staff will look after the pet (walk it, check for stress or sickness, make sure their food and water is refilled, etc).
When I talked to the Qatar office in Buenos Aires, they also require that I call them 3 days prior to the flight to reserve the space in the cargo of the plane. I know the travel time is over a day, but I am not worried since I know that someone else has travelled the same route with a cat a few years back and everything was fine.