Alright, look. This was published in the Proceedings of the National Academy of Sciences. Peer review isn't flawless, but it did pass it, and you've admitted that you don't know enough statistics to bring up any methodological issues--the most you can do is note that the effect is small .
I can easily question the methodology and claims which is what I did, what I can't do is read the finer results and regression statistics which may or may not matter given the basic idea behind the thesis is on shaky grounds.
On the purely statistic side I can try and make tentative question about whether they actually account for population density(to me it seems they corrected for raw population figures), why the R^2 of the trust-inquisitional density correlation and some other correlations is so low.
Another argument is if it really makes sense to exclude the 14-15 tribunal cities just because of potential bias when the argument is that the inquisition instilled fear and you'd think being near of where people were sentenced and maybe executed(I imagine people weren't executed back in their home village but maybe I'm wrong) would make those cities strongly subservient on this front.
I'm going to suggest that you might want to seriously consider that several centuries worth of religious persecution that only ended a couple centuries ago may have effects in the present day.
The authors clearly set up to prove their beliefs and this small effect what they ended up with, clearly it doesn't seem to have an actual large effect even if we say there are no issues with it.
But there are many factors that could make the analysis even more shaky, for example their GDP per capita figures derive from "nightlight" because they don't actually have GDP data for 3/4 of the municipalities.
By all means, produce evidence demonstrating that the scale of movement was so massive as to make this a nonsensical argument,
Well we would need to define what massive means first, the average modern municipality has an area of 60-70km2 and would have had a population of 1000-2000 people so at this point even marital ties or small distance migration between 2 municipalities, one that had a larger amount of trials and one that had a smaller amount, would muddy the results which makes sense considering how patchy the distribution is, also to consider is the cumulative impact of plagues, famine, coastal raids and warfare in the 1500-1800 period and then urbanization and industrialization.
keeping in mind that even if a lot of people, total, move around, if not enough move around at once they and their offspring will end up acculturating.
I don't see the logic in people quickly forgetting or starting to care about the inquisition just because it happens or doesn't happen in the immediate vicinity or that the supposed cultural transmission is this immediate and this complete.
If the inquisition was so effective in instigating fear in the wide population through relatively few trials(On average 1 trial per 10000 people a year? Or 1 per 5000 during the peak period in the mid 16th century?), it's weird that we are assuming that it only worked on a temporal axis for generations and not spatially beyond 10km from your location.