Do sound characteristics matter when moving from midfi to hifi?

Good question. But I would look at it differently. First, sound is tailored by high end companies very carefully. While consumer products are designed around cheap subcomponents and lots of features to put in the marketing material. So, the sound in high end equipment is carefully crafted to have a house sound. MacIntosh, Boulder, Luxman, and Audio Research all have house sounds. Then they refine these sounds the higher up the ladder of their products to sound better of that flavor.

Second is you will need to match your tastes to the company that produces the sound that matches your taste. So, for instance if I liked just rock and have my chest feel the impact of kick drums... I would pick MacIntosh and B&W. But if I was into Jazz and classical I would pick something completely different.

It took me a long time to learn what I wanted, and in the process changes what I liked. So, first order research that I would recommend is to listen to music. The kind you like, live. For me I went to acoustic jazz clubs and the symphony. That is because I didn’t just like rock, but all kinds of music. The best thing to do is to set your objectives on fidelity to the real thing, and all electronic music is unverifiable. So, learn what acoustic music should sound like... unless you only like Rock or only electronic.

The issue with just looking for what sound good is that while it is easy. To just go for detail and slam... as they are easy to sense. They are like the salt and fat in junk food. You can end up with a sparkling system that kicks ass, but has completely lost the emotional connection, the thing the draws you to live music. You can spend hundreds of thousands of dollars on a "sound spectacular" system, but that you can get bored with after listening for 45 minutes because it has no soul. While a musical system can captivate you for hours. So, before you buy, get out and listen to a real piano... jazz club, symphony. Then do research and listen to equipment.

I made a jump in audiophilia around 25 years ago similar to what the OP is about to do.

  I disagree about changing everything at once in order to note differences.  The issue there will be identifying which change did what.  I would do one category at a time and give at least a month of listening before moving on to something else.  Usually the first place to start is with speakers because once that is done right then it’s easier to assess other changes.  

Hypothetically, if going from Sony, Marantz, Schiit am I going to find something that I don’t like about Pass Labs, Conrad Johnson, Coda?

Oh hell yes. :)

I think that our industry pushes you to upgrade based on perceived value ($$$) instead of your hearing. 

Instead of asking yourself what is better according to  some rag, go out and listen and ask "what is this sound worth to me?" You end up with a better balanced view of not just what the value of equipment is but who YOU are as a listener and what YOU enjoy.  Of course I have my brand darlings, but they are MY darlings.  If you hear them and you don't like them, what difference does it make to you what I like, recommend or have waxed romantic about here? Same for reviews and the brand machines out there. 

Audiophiles who have given up upgrading for the sake of upgrading and have realized what they like and what sounds good for themselves end up happier and wealthy. :)

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.