Trying to bring cremated remains, or cremains, with you on an airplane can be a touchy process. The Transportation Security Administration (TSA) does try to treat cremains respectfully, but you have to do your job to ensure that the cremains are identifiable and scannable. Doing so will make your trip go much more smoothly.
Cremation urns can be quite beautiful, but many are made of metal or lined with lead, both of which will screw up the x-ray process. Those containers will return a vague, opaque image that will have the TSA refusing to let you take the package with you. Always transfer the cremains into a travel- and x-ray-friendly container. Wood, plastic, and special ceramics that don't contain metal or lead can work.
It is possible to check cremains with your luggage if you want on some airlines. If you prefer that, look for airlines with policies that spell out that you can check them. Otherwise, you'll have to keep the cremains with you as part of your carryons. If you are flying an airline that does not have a policy regarding checking cremains, you're better off keeping the cremains with you.
Another issue is where you're flying to. In the United States, you can usually keep cremains with you through the entire flight. But if you're traveling out of the country, the destination country might have strict rules about who can bring in cremains and who can receive them. Never bring cremains with you unless you've double-checked what the destination country's regulations are. You should be able to do this by contacting a cremation association here in the United States as well as contacting an association in the destination country.
Even the Urn Needs Identification
Don't forget identification for the urn. You'll need two sets, one consisting of a death certificate, cremation certificate, or other paperwork that shows these really are related to a funeral and cremation, and a second set to attach to the urn itself. That paperwork tells the TSA that this is a container that needs to be handled carefully and not opened. The TSA is not allowed to open cremation urns at all.
If you have any questions, talk to the crematorium about travel restrictions. If the staff there is unable to give you detailed information, they should at least be able to tell you where to go for answers.