The concept of "logical rudeness" (which I'm pretty sure I first found here, HT) is one that I should write more about, one of these days. One develops a sense of the flow of discourse, the give and take of argument. It's possible to do things that completely derail that flow of discourse without shouting or swearing. These may not be considered offenses against politeness, as our so-called "civilization" defines that term. But they are offenses against the cooperative exchange of arguments, or even the rules of engagement with the loyal opposition. They are logically rude.
Suppose, for example, that you're defending X by appealing to Y, and when I seem to be making headway on arguing against Y, you suddenly switch (without having made any concessions) to arguing that it doesn't matter if ~Y because Z still supports X; and when I seem to be making headway on arguing against Z, you suddenly switch to saying that it doesn't matter if ~Z because Y still supports X. This is an example from an actual conversation, with X = "It's okay for me to claim that I'm going to build AGI in five years yet not put any effort into Friendly AI", Y = "All AIs are automatically ethical", and Z = "Friendly AI is clearly too hard since SIAI hasn't solved it yet".
Even if you never scream or shout, this kind of behavior is rather frustrating for the one who has to talk to you. If we are ever to perform the nigh-impossible task of actually updating on the evidence, we ought to acknowledge when we take a hit; the loyal opposition has earned that much from us, surely, even if we haven't yet conceded. If the one is reluctant to take a single hit, let them further defend the point. Swapping in a new argument? That's frustrating. Swapping back and forth? That's downright logically rude, even if you never raise your voice or interrupt.
The key metaphor is flow. Consider the notion of "semantic stopsigns", words that halt thought. A stop sign is something that happens within the flow of traffic. Swapping back and forth between arguments might seem merely frustrating, or rude, if you take the arguments at face value - if you stay on the object level. If you jump back a level of abstraction and try to sense the flow of traffic, and imagine what sort of traffic signal this corresponds to... well, you wouldn't want to run into a traffic signal like that.
Another form of argumentus interruptus is when the other suddenly weakens their claim, without acknowledging the weakening as a concession. Say, you start out by making very strong claims about a God that answers prayers; but when pressed, you retreat back to talking about an impersonal beauty of the universe, without admitting that anything's changed. If you equivocated back and forth between the two definitions, you would be committing an outright logical fallacy - but even if you don't do so, sticking out your neck, and then quickly withdrawing it before anyone can chop it off, is frustrating; it lures someone into writing careful refutations which you then dance back from with a smile; it is logically rude. In the traffic metaphor, it's like offering someone a green light that turns yellow after half a second and leads into a dead end.
So, for example, I'm frustrated if I deal with someone who starts out by making vigorous, contestable, argument-worthy claims implying that the Singularity Institute's mission is unnecessary, impossible, futile, or misguided, and then tries to dance back by saying, "But I still think that what you're doing has a 10% chance of being necessary, which is enough to justify funding your project." Okay, but I'm not arguing with you because I'm worried about my funding getting chopped off, I'm arguing with you because I don't think that 10% is the right number. You said something that was worth arguing with, and then responded by disengaging when I pressed the point; and if I go on contesting the 10% figure, you are somewhat injured, and repeat that you think that what I'm doing is important. And not only is the 10% number still worth contesting, but you originally seemed to be coming on a bit more strongly than that, before you named a weaker-sounding number... It might not be an outright logical fallacy - not until you equivocate between strong claims and weak defenses in the course of the same argument - but it still feels a little frustrating over on the receiving end.
I try not to do this myself. I can't say that arguing with me will always be an enjoyable experience, but I at least endeavor not to be logically rude to the loyal opposition. I stick my neck out so that it can be chopped off if I'm wrong, and when I stick my neck out it stays stuck out, and if I have to withdraw it I'll do so as a visible concession. I may parry - and because I'm human, I may even parry when I shouldn't - but I at least endeavor not to dodge. Where I plant my standard, I have sent an invitation to capture that banner; and I'll stand by that invitation. It's hard enough to count up the balance of arguments without adding fancy dance footwork on top of that.
An awful lot of how people fail at changing their mind seems to have something to do with changing the subject. It might be difficult to point to an outright logical fallacy, but if we have community standards on logical rudeness, we may be able to organize our cognitive traffic a bit less frustratingly.
Added: Checking my notes reminds me to include offering a non-true rejection as a form of logical rudeness. This is where you offer up a reason that isn't really your most important reason, so that, if it's defeated, you'll just switch to something else (which still won't be your most important reason). This is a distinct form of failure from switching Y->Z->Y, but it's also frustrating to deal with; not a logical fallacy outright, but a form of logical rudeness. If someone else is going to the trouble to argue with you, then you should offer up your most important reason for rejection first - something that will make a serious dent in your rejection, if cast down - so that they aren't wasting their time.