Technically not. Dublin regulations should prevent asylum shopping, so if Finland processed (or started to process; having scanned fingerprints and put the person into the EURODAC database) their asylum requests, Finland becomes a country responsible for their asylum and protection and they should not be able to obtain asylum elsewhere. As in, other countries have the right to answer refugee's future requests with: "okay, but you were already in Finland, you go there." However, while third countries are not obliged to provide asylum for someone who already has "his" country, they can do so voluntarily - so for example Germany can take "Finnish" refugees, but it does not have to (they would have to take only "German" refugees).
That is the theory. However, noone cares about actual regulations, nothing works and everything is possible.
That is the theory. However, noone cares about actual regulations, nothing works and everything is possible.