I have just brought my cat and dog to Doha from the UK. Each country has very different rules for exportation - so, from what I gather, there is no one company who handle the entire journey from export through to import.

I used a logistics company in the UK, through DEFRA (department for environment, food and rural affairs in UK. I imagine there will be similar in both the US and Australia) to arrange the necessary things for exporting my pets from Britain. I arranged a health certificate and all the required vaccinations for both animals through my local vet under the guidance of DEFRA ruling. I then used Doha Vets for the import license and for picking them both up from the airport. You will need to speak to your vet in your current country of residence and then use either QPR or Doha Vets (you can google both) to sort out paperwork this end. If you use a logistics company, they will book the pets on to an airline (mine flew on the same flight as me) and liaise closely with you to help you sort everything out your end. If you then speak to QPR or Doha vets they will arrange import and everything this end and will liaise with your chosen logisitcs comany as well to ensure all required documentation is complete.

Hope that helps a little bit. Good luck - it seems quite daunting but when it come down to it it is fairly straightforward.