Across my life I have encountered this numerous times. One recent example:
If you don't think those are a crime, you are not adult enough or logical enough to have a conversation with me.
While this clearly includes ad hominem on fictional (or at least subjective) grounds, I'm asking about the first part of it, which presupposes that I'm too young or lacking logic only because I disagree with them.
What is this trick called?
Suppose someone says something like the following as the OP quoted:
If you don't think those are a crime, you are not adult enough or logical enough to have a conversation with me.
This is a threat by the speaker to punish the one who doesn't agree with what is called "the silent treatment". Wikipedia describes the silent treatment as follows:
Silent treatment (often referred to as the silent treatment) is refusal to communicate verbally with someone who desires the communication. It may range from just sulking to malevolent abusive controlling behaviour. It may be a passive-aggressive form of emotional abuse in which displeasure, disapproval and contempt is exhibited through nonverbal gestures while maintaining verbal silence. Clinical psychologist Harriet Braiker identifies it as a form of manipulative punishment.
It is an attempt to convince the other person to agree without using an argument.
Reference
Wikipedia, "Silent treatment" https://en.wikipedia.org/wiki/Silent_treatment
Its not really a trick at all. They are stating that "You disagree with me" is sufficient to prove "You are not {property} enough to continue this discussion with me."
There is a small trick, in the choice of words. As written, the focus of the last sentence is you, not the speaker. It's "your fault" that the conversation cannot continue. Such a statement would not be considered logical unless it was backed up by other facts.
However, there is also an equivalent construction which is "Let X be the class of people that disagree with me on this topic. You are in class X. I am not interested in discussing the topic with people in class X." This is a very logical construction. The key sentence, however, focuses on the speaker, not you. The sentence became an "I am..." sentence. As such, it is hardly refutable. After all, one does get to make such decisions about who one talks to.
If you blur these two constructions, it's possible to have the irrefutability of the second case mixed with the "you're at fault" voicing of the first. Such a construction is very damning, and can feel like an Ad Hominiem attack.
One approach to dealing with this is to force them to take a stand -- pick the first construction or the second construction. Ask "Why can't someone have such a discussion with you if they have a dissenting opinion?" By using the 3rd person "someone" rather than the 1st person "me," you give them a chance to make a choice. That makes it very hard for them to maintain the irrefutaility associated with saying something about themselves and simultaniously put the fault on you.
In any case, there is a logical construction which could be used here if you wished. Somewhere buried in the particular topic of disagreement is something which the speaker believe must be true in any consistent logical position. They may believe that any other position will yield an inconsistency, and the associated explosion of truths that comes from a logical consistency. For example, it is not uncommon for people's beliefs around God to be in a form "any answer besides mine must result in inconsistency". This can create issues, for instance, if a Christian and a Hindu engage in such debate. The concepts of God from those religions are sufficiently different that it can be easy to simply not understand that the other position is plausible.
However, finding and proving a logical inconsistency in someone else's mind is excruciatingly difficult. They may not be willing to engage in that effort. On the other hand, if they are wrong, then they need to remove an axiom from their logic ("All consistent belief structures have X" is no longer proven true). Doing that is also excruciatingly difficult. They may simply not be interested in expending such energy at this time.
