Do countries exchange DNA profiles from crime scenes with each other to check against national databases?
most likely would be done through interpol, if it's thought the perp may be operating in more than one country... an exception would be canada and the us, where there may be more a direct sharing of info...
I know that a few large organisations - the EU as the best example - do exactly that. Although they're trying to standardise legal systems, so I don't think think it's done between every country. Personal information exchange requires official authorisation at a government level - data of large transactions and extradition rights, for example, between the US and the UK.
Well I'm not sure if your familiar with CODIS but when a person is convicted of a felony their database is put into the computers. They do share profiles so I guess if there is an unsolved crime they probably will run through the database and see if there is a DNA match and if the person has left the country they probably will be red flag too.
There was a case a few years back where a torso was found in the thames. After doing some tests on the body they found that the body part had originated from Africa. I found a quick page on Wikipedia about it which doesn't contain much details, but you may want to look into similar cases to see how they have been solved to see if there have been cases where information has been shared between countries.
In general, law enforcement in different countries do not have direct access to the databases of other countries. There have been some initiatives to globalize this sharing, but as it currently stands, requests have to be filed for individual searches in most cases, with the usual delays whenever bureaucracy is involved.
I don't believe there is any intergovernmental DNA sharing system. However if the police have DNA at a crime seen and reason to believe that DNA comes from a foreign national who's DNA is on record with his/her government, they can ask for a sample of that DNA to run a restriction digest against the crime seen DNA.