There are multiple cases have been reported. I cannot be sure about the others, but in my case, I have exported to FBX, Imported to Blender and got texture mismatch. Essentially it was caused by pulling the texture from a wrong export in my case. It was an OBJ export folder, which I exported a few days ago with older Base.
I cannot replicate this bug anymore though. It is obviously a bug/glitch, where some texture path is hardcoded somewhere and then used randomly in consecutive exports. Please see the screenshot and this post for little more details: https://forum.reallusion.com/FindPost454898.aspx
I showed that in the video below :)
But in case it is unsolvable, I suggest to enforce export into a separate folder (or at least create an option with check box). There was such an option in CC 2.0
I have partly solved the problem. Blender takes the maps from the first folder that is in the same directory as the FBX/OBJ file, regardless of its name. So you need to create a new folder and put both the FBX and its texture folder inside, and no other folders. Then Blender will pull the right files right from start. But I have got a new problem, which I will open a new ticket.
I have just retested, CC3.3+ neutral male will export with right textures. But when loading the character in CC3, it will not load with boxers even when you apply all changes. The reflection map problem happens when I load CC3 Base Male and export it. I made sure to make a different name folder, and it was pulling the image from the correctly named folder. But the result is just mismatched. I guess that export settings of CC3 is wrong or export template. I'm quite surprised that such problem lasted so long without anyone asking about it. Male characters are quite unpopular or what.
Actually disregard the above comment. I think I narrowed the problem down.
Download the zip.
Unzip it and export as FBX with Blender settings and "Embed Textures" UNCHECKED any CC3+ character into the TEXTURE_TEST folder (at the same level as "A" folder).
Name export as Test, or something.
Now import that into Blender and enable materials. You should see "wrong texture" text on the character body.
Apparently it pulls the texture from folder which is 2 levels down A\B\C
I also posted that at the forum so people can try (hopefully it is not only me).