This is something I think of whenever a new writer happens for characters I love. Characters whom I've been following and following character journeys, developments, cast relations, how they think when shown, and charazations.
Many times if it's not the main series writer I find the character might be presented/shown not accurately or altered in some way not following what I feel is the one individual in the multiverse character's continuity. There are workarounds but some necessities has to be established.
I think for every example we present WE MUST state what character but more importantly what Earth version of the character we are writing about as for example if I was writing about 'Gwen Stacy' of '616' Earth she will NOT BE the 'Gwen Stacy' of 617, 1610, or 65 or any other versions and each is a independent character. The fate or what a character version does has no real importance/dominance to another version.
I enjoy Earth-65 Gwen Stacy/Spider-Woman but when she is written by another writer outside of her own series not written by Jason Latour then things might be presented as I don't see as 'accurate assumption' of that one character's continuity.
I felt this way doing 'Web Warriors' series as in that team Earth-65 Gwen Stacy/Spider-Woman was apart of that team. To note that if that wasn't Earth-65 Gwen Stacy version then I wouldn't be worried for the character's continuity representation and developments as she would be a fresh new version from scratch with no continuity to challenge thus a new individual character and just with her my hopes/assuitions of characterization desires to be shown and also willing to see the unexpected and hopefully well writing individual.
In Web Warriors that Gwen was often presented as 'experienced and one of the best'. However in her own series she makes TONS of mistakes, has her emotions cloud her judgments, suffers somewhat from PTSD, still not quite as experienced as the other Spider heroes of the Multiverse, seeks mentorship with Jessica Drew and by others around her, and falls on her face many times. Now if it was presented that the other Spiders saw her and based on their own self doubts but also rising others up in putting themselves down they saw her in a positive light meanwhile Gwen also has the same negative thoughts about herself and raises the others up in her own head equally then it not a matter of 'accuracy' but a matters of shown purposely inaccurate perspective in story and characterizations. A case of 'grass/hero is greener/better on the other side' mental state the Spiders all seen to suffer from as they see themselves as mess ups.
Based on info given to us about character versions I either come with a series of hopes and expectations based on continuity or just based on hopes of how a character might be portrayed. If it's a new character with no continuity yet made then I still have hopes and expectations but if the writer does something different the character presented to be could still be wonderful in ways I didn't consider or just works once everything comes together in story.
I'm feeling this worry for Gwen Poole of 616 having enjoyed her series but most of all her developments that made her a responsible hero and cares deeply for others beyond herself and connections and smarts/thinking that was shown. I feel worry for her upcoming West Coast Avengers as the writer wasn't her main series writer and I fear she will be presented just like how many who didn't read her series might present her which is inaccurate 99% of the time.
For the upcoming Spider-Verse movie that Gwen Stacy/Spider-Woman ISN'T Earth-65 version so changes to her differ from the ones I would of placed on the Earth-65 can happen as this upcoming Gwen has no continuity yet presented to us but what was given in the website bios and trailers (not liking what I read in her bios however so I'm hoping against that info or a good twist happening). That said I still have certain hopes based on my own perspectives to this new version. This Gwen might not be a drummer and might instead be a 'geek gamer girl' and extroverted and hyper and that would be very different from the introverted brooding Rock drummer Earth-65 version and for me that is FINE. What I do hope is seeing what shaped this character version into who she is, how her powers work if different from Earth-65 version, her thinking and how it works and influenced her life choices, and finally that she is inexperienced and falls on her face equally as a the guys helping to teach each other while also still learning from each other and not a 'pro'. A strong likable character I can relate to or at least if I disagree I can STILL see where the writers were coming from and other might latch onto.
Some of the examples posted by others such as the cartoon versions and movies isn't really a 'misrepresentation' as it's more of a 'representation' that we didn't like and fear might affect how we could see other versions past and future. It's a small but still a big difference. Writers making fresh new versions and Earths are free from the chains of continuity and must focus on making a world, characters, developments, and overall story that is fantastic even if it might different from the various versions sources. These versions we have hopes and expectations influence/based on past/present versions but anything goes.
Writers when writing for a 'established' version character I hope will put in the time to research and greatly read the past couple of trades of the main series they were in and other books for contitlity and best representations they can do. But the matters of how they see a 'established' individual character version and how I see the same 'established' individual character version still might differ.
We seen this happen on these fan forums where how two people see the VERY same character and interactions of what is happening in one issue can be very different. Same can happen for different writers and can be influenced or the thinking can go in different ways yet hold the same information gathered.
When writing in this very topic we must write more then stating a character was 'misrepresented' but MUST present the why and hows and give past info on that version that caused the feeling of conflitions of charazations we are complaining about due to disappointments.
If you write version 6872 Peter is different from 734 Peter then that isn't a good example as they are two different versions/character really. Peter of 6713 can only conflict with past info of 6713. You however can still say that there was no conflicts but still post disappointments of the overall results if you felt the movie,comic, or tv show failed in some way of your expectations or the writers tried something but missed or caused a plot hole that they missed/paradox or was lacking in showing how the individual character was influenced that they missed or didn't build upon enough.