It really depends though. I’m surprised by how many people are so certain this can only be a rider issue. It’s definitely worth considering but not the only likely cause. If it were really as easy as “ride them the same way to get the same results” you’d never see top level riders having any issues. Sometimes horses don’t get the memo 
My horse also often loses relaxation moving from the warm-up to the show ring and it’s definitely not a rider issue, he just gets anxious in new places. He starts off warm-up tense too, but the difference there is we have plenty of time to walk around and let him look at things and process before going to work. Then we move to the show ring, which is a new spot with new things to look at and be worried about, and we only have a minute or two to take it all in before we have to start our test. Leaving the group can also be a factor for a lot of horses so since OP notes her guy is herdbound in general that’s something I would look at closely.
For mine it’s just been a matter of exposure and building his trust in me over time. As he’s gotten out more his threshold has increased, his reactions are less dramatic and he takes less time to settle. We’re still not at a place where he can move to a new area and just go without missing a beat, but it’s getting there.
I’d also play around with how you spend your lap around the ring before going in and see if that helps at all. Some horses need to be put to work and distracted, but for others (mine) that just makes it worse. I like to walk on the buckle as time allows, just to take the pressure off and give him as much time to process as possible before picking him up. Ideally I’d like to have more time to get him working before turning down centerline but right now we really need that relaxation instead.
