For various reasons, I wanted (and got) a position that paid significantly less, and entailed a "demotion" in title, compared to what I was making with another employer - I expected it to be puzzling to the prospective employer why I would make the move, and it was. Compounding the issue, I'm in a field where people often fake-apply for positions, just to get offers they can use in negotiations with their existing employer... And the position was located in a place that is generally regarded as less appealing than where I was living at the time... All up, from their point of view, there was a huge risk that I was making a fake application, or that there was some invisible thing wrong with me to lead me to consider the move - or maybe that I was using them to relocate to the region, and would promptly leave for better pay once I had clocked a bit of local experience...
I used a combination of strategies to persuade them, but mainly I leaned heavily on substantive elements of the position that I couldn't do in my current role - whatever was new about the position, that was what I was keen to do. I /did/ say that money wasn't the most important thing to me (even ended up having a conversation somewhere along the way about being frugal in order to explain why...), and I cited family reasons for being interested in moving to the location where the new position was based.
I got the job. And, as it happens, I failed at the "downsizing" aspect of it - I'm a couple months in, and just got promoted back into a more senior role with a nice boost in pay (still a bit less than what I was making in my old job, but I'm in a lower cost of living environment here)... So, in the end, the trade-off wasn't as drastic as they were afraid it would need to be - I think if they had offered, upfront, what they've just offered, they would have been much less anxious about whether I was serious about the position...
I agree with others that you probably need to find positive reasons for actively wanting the role. While there are some employers who might not mind, most will want to feel wanted... They will take that as a proxy for "fit" and your likely commitment to the role.