I'm also finding it hard to describe the concept succinctly, which might mean that it's not well-formed, but I'll give it a shot anyway.

The concept I'm trying to put a word to is the rhetorical technique where a monologue starts on one subject, perhaps a subject that's already reasonable or agreed-upon to the listener, and, through a series of perfectly natural related tangents, somehow connects to a disparate assertion, especially when the goal is to convince the listener of that assertion. It sort of reminds me of the word game where you take one word, replace a letter to get another valid English word, and repeat this process until you get a different word, like COLD -> CORD -> WORD -> WARD -> WARM. Ideally, I'd like to find the verb or verb phrase form of this (something defined roughly like "moving rhetorically from point A to point B using natural extensions or tangents of previous subjects").

Taken to the level of parody, you might get The DaColbert Code or Fat Tony's bread analogy. Hamfisted attempts at this happen all the time in commercials, with monologues like "At $COMPANY, we know that what's important to you is $VALUE. That's why we're $PROMOTION."

Something like 'conjoin', 'conflate', 'imply', 'free associate', none of these seem to work quite correctly. The ideas I'm trying to express in this term include both the disparity of the beginning and end subjects and yet the overall lack of 'seam' or 'break' in the conversation -- each step is a natural outcropping of the previous part of the conversation. I would call this 'deploying a Ship of Theseus argument' but my understanding of the term is that a Ship of Theseus argument is an inflammatory accusation that deceptively redefines most or all of the words in that accusation to attempt to vilify benign behaviour; by contrast, all of the steps are sequential and out in the open, and the goal is not (necessarily) to deceive.

I'm also finding it hard to describe the concept succinctly, which might mean that it's not well-formed, but I'll give it a shot anyway.

The concept I'm trying to put a word to is the rhetorical technique where a monologue starts on one subject, perhaps a subject that's already reasonable or agreed-upon to the listener, and, through a series of perfectly natural related tangents, somehow connects to a disparate assertion, especially when the goal is to convince the listener of that assertion. It sort of reminds me of the word game where you take one word, replace a letter to get another valid English word, and repeat this process until you get a different word, like COLD -> CORD -> WORD -> WARD -> WARM. Ideally, I'd like to find the verb or verb phrase form of this (something defined roughly like "moving rhetorically from point A to point B using natural extensions or tangents of previous subjects").

Taken to the level of parody, you might get The DaColbert Code or Fat Tony's bread analogy. Hamfisted attempts at this happen all the time in commercials, with monologues like "At $COMPANY, we know that what's important to you is $VALUE. That's why we're $PROMOTION."

Something like 'conjoin', 'conflate', 'imply', 'free associate', none of these seem to work quite correctly. The ideas I'm trying to express in this term include both the disparity of the beginning and end subjects and yet the overall lack of 'seam' or 'break' in the conversation -- each step is a natural outcropping of the previous part of the conversation. I would call this 'deploying a Ship of Theseus argument' but my understanding of the term is that a Ship of Theseus argument is an inflammatory accusation that deceptively redefines most or all of the words in that accusation to attempt to vilify benign behaviour; by contrast, all of the steps are sequential and out in the open, and the goal is not (necessarily) to deceive.

EDIT 2023 JAN 30: In reading the comments and attempts at answers, I'm going to try and refine my question a little. I think the 'starting from a premise intended to be acceptable by all parties' is an important part of the term I'm searching for (and the connections definitely aren't tangents or nonsequiturs, but logically or conceptually connected in some way).

I placed a comment about this below, but for clarity's sake, the commercial monologue example attempts to find a value that the audience will have and then bridge the state of having that value towards a conclusion that on its face doesn't seem to have anything to do with that value. So a commercial could argue that $COMPANY knows you value family, and that's why they made $PRODUCT, which is so convenient that it will give you more time to spend with your family. "Buying a product" does not itself have anything to do with valuing family, but each step in the argument (buying a product -> convenience -> having additional free time -> spending more time with family -> valuing family) connects logically with the last, and the argument ultimately argues that it would be an alignment with your values to buy the product.

To be clear, this argument does not need to start on a value, but frequently does (the Fat Tony argument starts on an ethical question, for example, and a statement of fact would also be fine). The answer below about chain arguments is the closest so far, but seems to miss the part about starting at a place the audience would agree with.

In trying to describe this, I'm hoping I'm not just trying to define the entire topic of persuasive argument. o3o I think I'm aiming narrower than that.