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I’m not familiar with the Finnish (asylum) law, but this is what I found out based on my research:
If a citizen of a EU-country (like Finland) gets married to a non-EU citizen, then the foreigner will receive a residence permit (issued by the state that the EU citizen is a national of). The residence permit also enables him/her to work within the EU. Your fiancé will be allowed to move and work within the EU.
For further information about how the application procedure for a residence permit in Finland works, please check out the following link:
You don’t have to marry in Finland just because your fiancé applied for asylum there. In general marriages within EU will be acknowledged by every other member state. But it might be easier to marry in Finland to avoid bureaucratic hurdles. For example it could be difficult for him to travel to other countries without having a refugee passport. And you will have to register a marriage that has been done abroad at a Finnish register office before he can apply for a residence permit. See this website for example: http://www.migri.fi/moving_to_finland_to_be_with_a_family_member/filling_in_the_application/family_member_of_a_finnish_citizen/spouse/marriage_registered_partnership