Personally, I think the converts are like people who join any cult. They are adrift, "seeker" types who feel like misfits, for one reason or another, in contemporary society, and they are attracted to Islam because it takes total control of their lives, tells them everything they should do, down to the smallest detail, and brings them into a special, closed society where they feel important. In the case of Islam, the cult's devotion to violence and conquest also makes them feel powerful.
I wonder if the radicalized children of the immigrants feel that way also? They seem to be perpetual students who never grew up. Rather than blame themselves for being failures, they gravitate to radical Islam, that teaches them that they are the vanguard of the new world order (kind of like communism and Naziism). I think the welfare benefits European countries give such shiftless people contributes to the problem. You have radicalized, alienated young people who live on welfare money and spend their days being taught Jihad.