Some MFT cameras have for example a higher DR than some FF cameras of the past. My last word on this is that I think it's important to try and identify exactly what we're talking about so that we can if possible close any gaps that do appear by doing things differently when using different systems.
I don't think it's just dynamic range, depending on what you exactly mean by that, and it may well have an impact on what I, and others are seeing.However, I'm not sure if I can explain any better than the several posts I've already made about tonal and colour gradation, because I think this is the key difference for me.
You could see it in the film days when you moved from 35 to roll film to sheet film, and the same applies to sensor sizes, though I think it's complicated by pixel density and bit depth.
I used to be able to go through books of photographs, and pick out the photographs made on sheet film, (as opposed to roll film or 35mm) based purely on the colour/tonal gradation qualities of the images.
As I said in an earlier post, the difference is most noticeable, but not exclusively so, in areas of similar tone and colour. e.g water, skies, skin tones, etc.. With smaller sensors, tones/ colours change fairly abruptly, but as sensor sizes increase there is a smoother change between colours/tones.. This often has the effect of making the global contrast appear flatter, but local contrast higher. Giving a "soft" pop to the print, rather than a "harsh" pop to the print.
It's certainly more noticeable with some subject matter over others. but possibly the most glaring example for me was a photograph of a kestrel on a perch. One with a Nikon one V3 and the 70-300 lens, the other with a Nikon D7000 and the 28-200. The 1" sensor on the Nikon gave me a pin sharp, grain free A4 print, which I was really impressed with. Much better than I had expected from the 1" sensor.
But the print from the Nikon D7000 file, although not as sharp, had nuances of colour in the feathers that the Nikon V3 just didn't have. Feather areas that looked a homogenous brown in the V3 print had multiple gradations of brown in the D7000 print. Looking at the wo print at normal viewing distances, the D7000 one was clearly nicer to look at.
It's an experience I've repeated over and over again, and for me, almost all the time the larger sensor results have an obviously different look to them, which just looks "nicer". They are also easier to process. I used to spend a long time working on 1" and M43 files, trying to get them to good starting point .
As I said in an earlier post, I've just got a Fuji GFX 50s, and so far, it's showing a noticeable improvement in colour/tonal gradation over my Nikon Z8. Which is what I was hoping for, I have a specific project in mind for it.
Obviously, how important these difference are, depends on what you are doing, there are many reasons to buy one system over another
In terms of closing gaps, I'm not sure it 's possibl. I have found in the past that making custom camera profiles can help (using Lumariver) which in my case with Nikon cameras improved the tonal/colour gradations in the greens. I have also found that Capture One (regardless of camera), gives slightly better colour/tonal gradations and micro contrast control than Lightroom or DXO PL can manage, but both these things aren't really a substitute for a bigger sensor.