Look in the immigration website for immigrating with your pets , it can take up to 6 months (in some cases, not all) usually because of rabies - that's worst case scenario, many people only have their pets in for 30 days. During which time your pet will have to be quarantined, and it really isn't cheap. People do it, but be careful picking the airline you use to fly your dog over some aren't known for good care.
Animals are usually flown in cargo so many don't recommend flying during the summer and winter because if heat and cold extremes. I wish I could give you more specifics but I did not migrate with mine due to their advanced age and cost (mine were 10 and it would have cost - in Australia - about $7500 each for 6 months of quarantine) so they stayed with my parents, we'd been living on their farm for several years and they were quite at home there. Both only lived a couple more years.
Yes, it was heartbreaking but my Visa was approved and I had to move then or not at all. It was not something I'd ever do again. I'd wait, if I could.. rehoming elderly dogs isn't easy. And quarantine isn't easy, but it's like leaving them in a kennel for holidays. Perhaps you can visit often. Now, all that said, if you have all your ducks in a row then it should be quite smooth. Again, check the immigration website, there should be a section on migrating with your pets.