Do sound characteristics matter when moving from midfi to hifi?

You can absolutely upgrade from midfi to hifi tier and not like the result. There are matters of synergy and taste at play. Look at the used market! Lots of high end gear getting constantly shuffled around. Lots of mid and low tier gear too, and some of those sellers may be trying to reach for the next tier for satisfaction :)

However, higher tier components get to remove budget constraints in their design and implementation. This allows changes that matter....and some that probably don’t (cosmetics). But IF it was engineered well and voiced right and synergizes well with your system...it can be magic. If you get one of these components to elevate your system, it will probably hype you up to go buy more high-tier components...which you may experience mixed results with :P

There is truth in all of the above posts. There is singular path on this journey, there will likely be regressive, lateral and hopefully progressive moves, I've certainly experienced this. For me its been a long slow journey, I began by listening to lots of live concerts and audio systems at dealers and shows, In doing this I developed a reference for preferred presentation and how hifi differed from live sound, this not a planned thing, really didn't know a thing about high end audio at this point. Alongside this purchases of equipment, started off relatively low end, progressed through mid grade and on. Having heard my references for high end audio early in the game, and of course new ones continually popped up I only knew my own system wasn't up to the task which drove me to continually upgrade to the point where I now have the system I always strove to have. This was a decades long journey for me, I suppose today's audiophiles have it easier in some ways, harder in others. Easier in that so much knowledge to be gained via interwebs, harder in that who knows what info is good or applies to my situation and many fewer dealers, I spent a lot of time, learned a lot from dealers and reps at shows.

As for addressing your specific question, generally we're reaching for ever greater resolution/transparency in our systems, as others have mentioned this is likely to uncover weak link elsewhere which in turn means addressing that weak link, and this can go on and on. Recently I saw a youtube video from some audio show where a panel of 'experts' questioned the audience about their efforts to attain ever higher levels of resolution/transparency and their level of happiness with the results. Most raised their hands when asked if they were frustrated/unhappy with their present situation. What this says may seem pretty disheartening to many, I'll just say don't expect it to be easy, I've been pretty OCD over the years. On the other hand, who can say, every person unique, contentment may be found far more easily for some than others.

Meant to say NO singular path.

I agree that changing a single component in a system may, although not necessarily, noticeably alter the sound. For my first post-college system, I visited stereo stores in Anchorage. (During the pipeline construction years, lots of folks had some extra bucks, and there were several audio stores.) I couldn’t afford what I really wanted, but bought a Garrard turntable, an Onkyo receiver, and ADS 710L speakers. Being a buy-and-hold kind of guy, only several years later did I swap out the Onkyo for an Adcom integrated. The improvement was significant. More than several years later and after adding a Sony ES disc player, I traded the ADS speakers for Harbeth M30s (which I still use daily). Again, great improvement. Then I traded the Adcom for a Classe, even better yet. Then the Classe went kaput, and I bought a wyred4sound integrated, which I think is the best so far.

Now, I heard improvements but my choices haven’t been true high-end. But they have given my wife and I decades of pleasure. And I’ve enjoyed the process of making changes when I have. My hearing has diminished, and I don’t know whether my approach would give satisfactory results, given higher costs for audio equipment, especially speakers, which in theory are due for an upgrade. 

It is very hard to describe in words aural phenomenon, which is one reason why reviewers are constantly seeking better (and often more hyperbolic) adjectives. "Mid-Fi" and "high end" are gross labels that don’t tell us much: in the era of late 60s to early 70s you might more easily characterize a mid-fi system as one which consisted of a receiver, a pair of bookshelf type speakers and a modest turntable, compared to separate preamp, basic power amp and turntable that allowed the installation of a separate arm (often made by a different company). These days, some lower cost hi-fi equipment sounds worlds better than its equivalent from 50 years ago-- solid state is much better sounding, to my ears, than it was in the early ’70s.

It’s very hard to predict what a given set up will sound like in the abstract, leaving aside listener preferences. There’s the room to consider, and how the equipment is set up within a given space with its acoustic characteristics. Some of this is science and some is art. If I had to describe what a "high end" system would sound like in the abstract, compared to something lesser (and this isn’t necessarily measured by price or brand name) it would be: a more natural "flow" to the music without sounding "reproduced" or "played at you,"; a slowing down of the proceedings--which may be partly psychological, where there is more "space" between the instruments, both in terms of timing and physical placement in the presentation; front to back "depth" as well as the harmonic decay "envelope" (from attack to the gradual diminishment of overtones that linger and fade) and an ability to "decode" information in a way that isn’t analytical or clinical sounding, but makes your brain "work less" to imagine the illusion of a real performance. If you know what a real instrument sounds like, say, the piano, you need a recording that has effectively captured it (not easy in my experience) to hear what a good system can do in reproducing it effectively, from the "growl" of the lower registers to the sparkle and ethereal quality of the upper notes and the harmonics that can linger as each new note is struck.

Each of us has an imagined "ideal" for good sound. I’ve heard a lot of systems, big and small, in homes and studios, over the course of more than five decades. Some of the big expensive systems fail and other more modest equipment just "gels" in the room, producing a convincing illusion. Part of this too is that you are evaluating a "system," not just a series of components. 

When I got serious about this hobby in the early ’70s- the Quad loudspeaker combined with tubes and a good turntable was eerie. It has considerable limitations in terms of bandwidth, dynamics and the ability to play "loud." But it is still a reference for midrange, which is where most of the action is. (I have a vintage system set up that replicates what I was running in the mid-70s that is based on the Quad Loudspeaker).

By contrast, my main system can play louder, deeper, has a similar "see through" midrange and does not suffer from the some of the limitations of the vintage system. Yet both are musical and very natural sounding.

I had an industry person here a year ago who wanted to listen to some test pressings of a sound track and kept asking me to "turn it up." He was listening for nits-- although my main system is more revealing than the vintage one (the latter being somewhat forgiving), neither system is "forensic" in its sound. He would have been far better off in a studio with big JBLs being driven by a lot of power (I use single ended triodes), to hear these pressings as if under a microscope. That’s not what I’m chasing; instead, it is about a quality of realism of instruments in the room.

Since I have two dedicated rooms here, the problem is largely one of space and scale. There is only so much I can crank the big system before it overloads the room. Thus, I cannot produce the scale of a large orchestra in a 2,800 seat auditorium, though I can produce a very convincing illusion of one in my room, given its size (roughly 31 x 14, narrowing like a horn mouth toward the front wall).

The real measure of this is going to be your ears and your listening experience. Get seat time; hear some top systems and what is possible. Work from there as a reference point, recognizing that sonic memory is a tricky thing.