any one reading this… Critically important to close unreal engine before attempting to move the files. Then move the files from the browser to the project in browser. If the project is not open it wont attempt to import them. For some reason it cant see its own file type. Nothing can see its file type for that matter. aka Gimp or something cant open a Uasset texture with zero changes to it.
2022 and it not solved yet? 
References of files still extinct after copying to a content folder
2023 and not solved yet 
after almost 9 years there is no real solution for this 
been trying for hours to import uasset files that i have and still no progress
Here is a video that covers how to move uassets into projects. How To Import .uasset Files Into Unreal Engine 4/5 (Tutorial) - YouTube
Guys I got it to work after 10 minutes. Like what previous people said, you can only migrate from uassets from a previous version to a newer version. Make sure Unreal is closed*, then drag and drop uasset into a folder in the content folder. Then boot it up and it should work.
Just adding to this fossil of a thread, but getting scathingly specific for the people still having issues. I purchased an asset off of Gumroad from Humble Bundle and got a download of some assets which were .uassets, you need to CLOSE UNREAL AND COPY + PASTE those into the direct file path for your project. So D/Documents/UnrealProjectFiles/GameName/VFX, etc. Once you reopen they’ll be right where you put them, now completely usable.
Mate, please stop giving wrong information, if you use this method you will lose all dependencies and hooks if you copy more than one uasset at a time. if you copy a simple cube you better make it from scratch, jeez.
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I have auto save .uasset that saved 10- mins ago will not go into my project that crashed why please help me tell me why? not by any method spoke of in this article it just does not show up if I try to important ue acts like my ex-wife acts around me at a restaurant. WHY WHY its hours of work why its the only thing broke I have 6 auto saves of this file
and nothing nothing, nothing ??? Please tell me why?
I know its late but for any new comers like myself the solution is to migrate or copy paste uasset files from projects using older version or same version of unreal engine to the project of your choice upgrading in this way is ok after that all you have to do is (if you didnt migrate assets) close the project and reopen it (might need to delete intermediate, binay, saved and DerivedDataCache folders if you do this you need to right click on the uproject file and generate visual studio project files) as for downgrading from newer version of unreal to older your best shot and maybe only option is to copy from blueprint editor to notepad and to the new projects blueprint editor