even if he claims refugee in Canada, and u sponsor him inland in Canada, his refugee file could be completed before your sponsorship is and there is still a chance he could be removed IF the refugee is completed and denied before the sponsorship.
They have sped up the refugee process significantly.
for example a friend of ours claimed refugee in January 2012 and had his hearing in june 2012 (5 months later) at this point he wanted to leave Canada because if you become a failed refugee its harder to return later, so he withdrew his refugee at the hearing and left Canada)
But im saying this because my husband was a refugee and he applied in 2009 and didn't get his hearing until 2011, and than didn't get his removal until 2012. If refugee applications took this long still, it wouldn't be much of a problem, but they have recently changed the process and made it a lot faster to finalize refugees.
I did hear of a lady, who sponsored her husband to Canada from cuba, and a month after he arrived he flew to the USA dropped his PR status and gained asylum in USA. BUT he is from Cuba and its a whole different ballpark for Cuba because they automatically get refugee in the usa if they make it to American soil. For other countries its different as Scylla is saying they may not allow him to do that because of the safe third country agreement.
I would suggest speaking to an immigration consultant or lawyer before you go ahead with anything, just to be sure of your options