I moved from Canon. I bought an X-T1 about 2 years ago as a travel & fun camera, with the 17-55 f2.8-f4, and the 35mm f1.4.
My main interests are wildlife, motorsport, and landscape. My bodies consisted of a 6D, 7D & a 7D2. My lenses included a range of L glass 17-40, 24-105, 70-200 f2.8, 300 f4 plus 50 & 85 f1.8 primes, Sigma 10-20 and 150-600 Sport; so quite a lot of kit.
I tended to take the Fuji out almost every day, and when the X-T2 came out, I bought that, with the grip and then the 100-400 with 1.4TC for wildlife. A this point I realised that I just wasn't using the Canon gear. The X-T2 was delivering what I needed for wildlife, and the landscapes were great too. I have since added a number of lenses to the Fuji kit, and all the Canon gear has gone.
Do I miss it? Well, up until the last firmware update, the 7D2 was better with moving subjects, but since then I don't have an issue. Yesterday I was walking around a reserve with the T2 & 100-400 on an R strap and my other T2 with the 80mm f2.8 & 10-24 in a rucksack on my back without a problem (and I do have a back problem). Migrating to Fuji has meant that I take my camera out more often, I enjoy using it more, that has resulted in me taking more (and better) images.
However, there is a learning curve, the Fuji menus take a bit of getting used to, and some of the options are (sort of) hidden. I had to learn how to PP the raw files, but that was fairly straight forward. I do like the idea that I now have a pair of bodies, both of which get regular firmware updates (that doesn't happen with Canikon) with a good range of lenses (the glass is great by the way). It's lighter than the equivalent DSLR, I can get a body & 2 lenses into a very small camera bag, or even a normal rucksack.
Go to a store and try one, I would suggest the T2, the H1 is still a bit new and unless you intend on using the longer heavy lenses, and use a lot of video, the H1 is probably not going to give you much more, and it's bigger/heavier. T2 has dual SD slots, and I've shot birds in flight & a wedding on them - what's not to like?