The green card does not give you comprehensive cover abroad. What you have to be given is the minimum cover required for the country you are visiting which will always be a type of third party cover for third party injuries and may include third party damage. Any more than that is between you and your insurer, some companies will give you comprehensive cover and others charge an additional premium so you need to check this out before going abroad.
As to long term cover abroad this varies between companies and you need to speak to your insurer to see what they are prepared to do, the larger companies with offices abroad may give you a better deal. Most foreign countries have a higher accident rate than us and therefore higher premiums, so expect to pay more.