I did LIM->MIA with a 1 year-old beagle on American Airlines in August.
1. Check what your temperature will be at your arrival airport at the time of arrival and for several hours after. I know AA will refuse the pet if the temperature at landing will be in excess of (I think) 80 F. I was fortunate that the flight was overnight, otherwise I would have had to choose a different airport farther north or wait until December.
2. Try to call and set up some type of reservation for the pet beforehand. They may say they don't do that, but at least ask a couple of different people, including a United representative in Lima. And regardless of that, arrive early. The reason being that some airlines have a maximum number of animals that they will take on a flight, so don't let yourself get turned away because other passengers have already checked in with dogs.
3. Make sure you have an up to date rabies vaccination and a certificate of health obtained no longer than (I think) 7 days before travel. The US customs agent can ALSO technically ask you for a certification that the dog is free from screwworm, although they didn't ask me that. The vet in Lima should be able to give you that though. The United representatives in Lima should be able to confirm additional documents needed.
4. Clearly mark the kennel as "Live Animal / Animal Vivo". Just print out some pages on computer paper in large font and tape them to the top and side.
5. Water dish is mandatory, as well as a bag of food taped to the side. Soft bedding material should be added, but nothing they can choke on.
6. Most vets recommend against using sedatives. Most dogs will just go to sleep in the dark environment anyway.
7. You may have the right size kennel but there may be more to it. There are requirements about the number of slits for breathing, how it is assembled (screws vs. clamps), the material it's made out of, no wheels, how the door is made, etc. Look into all of this in detail for your airline.
At the end, everything went well for us. Our dog was brought out to the baggage claim area in Miami airport soon after arrival, crying and confused but otherwise in good health, and she was very quickly her old self again. Because I had the health and rabies certificate, there was no problem at customs.