Sophistry and rhetoric are methods of dialogue associated with ancient Greeks like Socrates and Plato, wherein one engages in dialogue with the express purpose of making the other person contradict himself and seem foolish. I believe that those who are prepared for proselytizing are trained in a system in dialogue that is more-or-less the same as sophistry.
The person intended to be proselytized is led to imagine that he is engaging in an open and casual conversation about the comparative validity of his own religion, and that of the proselytizer. However, it is not "just" a conversation because the proselytizer has the deadly-serious agenda of converting the other person, while the other person usually does not have any agenda for being in the conversation. Also, the proselytizer may have back-up persons such as priest who brief and debrief him about such conversations for hours, and prepare him to hold various conversations or even to lure him to the temple/mosque/church or other place of worship of the proselytizer/converter, and into the company of more converters who may play various roles in the convertee's social life, such as accuser, sympathizer, good-cop-bad-cop, devil's advocate, girlfriend, boyfriend, etc.
The thrust of the proselytizer's conversations with the intended convertee is to seem utterly logical, tolerant and compassionate, so as not to provoke any defensiveness on his part. A key element is to induce him to relate to the proselytizer, and share negative confidences about his family, friends, social setup and co-religionists.
As a result of a series of interactions, the intended convertee often feels a sense of euphoria that he is surrounded by a bunch of new friends who understand him and are genuinely interested in the problems of his life, his emotional angst, etc., and give him the importance that he feels that he deserves. He contrasts this with his family members and his older friends, who seem relatively indifferent, uncaring and dismissive, and he feels driven towards the decision that his future happiness lies in proving himself loyal to his new friends by embracing their faith. Due to his frequent interactions with the proselytizers, the intended convertee may actually feel guilty for postponing his conversion to the new religion, which seems to be eagerly waiting with open arms. His old religion, by contrast, appears utterly lifeless and nondescript, and his family ties (which may break due to his conversion) don't seem worth maintaining at this point of his life.
A great deal of the proselytizer's conversation may be aimed at fomenting a sense of grievance against the convertee's co-religionists, and a spirit of discontentment and rebellion. "Show them that you are willing to part company with all that is untrue and unworthy in your life," the proselytizers seem to say.
If the intended convertee happens to be undergoing a bad patch in his life, or a transition phase such as adolescent rebellion, midlife crisis, joblessness, discontentment with marriage, etc, then the opportunity to 'make a clean break' with his life may seem too good to pass up. The possibility of being enabled to remarry or restructure his business, or find employment within the new community's network, is seductive.
Converting seems like the only decent, rational and beneficial thing to do. Not converting threatens his newfound friendships, happiness and progress in his life.
[My answer is intended to provoke discussion. It is illustrative of the environment set up by proselytizers, rather than the actual arguments themselves. This is admittedly not a classification of the kinds of arguments used by proselytizers, but it hopefully sets the stage for examining and classifying them.]