My GUESS, is that they were indeed initially seeking asylum. recall that the purpose of asylum is safety and freedom from religious and political persecution.
Many of the Burmese refugees hope that things will settle down, that their millitant government will be overthrown or drawn into line so that they can return home in safety. In the context of true asylum, they are not fleeing because they WANT to leave their country, but because they have genuine concerns for their lives but hope to return.
The fact that several thousand choose which camp they settle in is reflected by this theory. Some of the camps offer an education equivilent to the Qld (and Now Australian) Curriculum. The goal of which is to attain the skills and qualifications so that they can return to, rebuild and reestablish their home communities.
There are currently at least 600 odd students who have completed their education to a grade 12 Aust standard in these camps, however there are no facilities or resources for Teriary Education. There are however some trades available, and whilst the skills are relevant and applicable, may not be "internationally recognized", but certainly capable of providing the skills their current community needs to assist aide workers with sustainability.
For 480 of those Australian Entry Qualified Tertiary students, there are private sponsors and places in Australian Universities, however red tape (specifically their lack of papers) has so far thwarted attempts to extend the offer. In short, we have no legal way of getting them here. The places are offered on the understanding that once the degree is completed that they will return to the camp or their country (if safe to do so) to inject their skills back into their community with the goal of rebuilding. They accept that they will never be granted any status that allows them to remain in Australia, now or in the future.
My GUESS is that during those 10 years where they were safe, they got tired of waiting (The situation in Burma has been prevalent for over 60 years now) and either heard about "a better life" or found out how to get to the transport required to attempt the journey. I believe that there are very long waiting lists even for those who do "find out how", so it may also have been a case of them having to "wait their turn" for available transport.
Some people can go their whole lives and never really live for a single minute.