For any air travel, the weight of the case becomes an issue. In general, airlines charge a premium over 50 lbs, in addition to the charge for shipping a bike. A hard shell case is in the 30 lb range. There are several cardboard ones in the 20 lb or less range. When I shipped my Atlantis to Europe, I wound up stripping off the racks and carrying some parts (pedals, cassette, ??) in luggage to get under the 50 lbs limit with my hardshell case.
If you are considering a specific airline, check their regulations. Some of the European carriers will accept a bike as "roll on" with no boxing req'd. Other airlines require boxing. There is no standard. The cost even with the same airline can vary considerably, i.e., free on one leg & US$300 on the return.
dougP