This actually happened with me unfortunately.
I went out with this girl from Morocco for about a year and at the end she was continuously pleading with me to get married -- I had told her since day one I wouldnt be able to marry her because I would be marrying a Pakistani girl thorugh an arranged marraige and that would not change....she was aware of this and was cool with it....then at the end all hell broke lose...lots of drama, tears, threats of killing herself and what not....it was pretty screwed up...almost like a Bollywood movie....
Anyway, point being...No. I wouldnt. Not just because the person is pleading or 'begging'...but more because you should never have to go that far to find a partner. If you're seriously that keen to get married because 'time is running out' or whatever....then you have issues.