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Use Dependency Walker or it's successor Dependencies to find out what dlls the Qt mysql plugin needs. Most likely it's libmysql.dll and it's dependent dlls (openssl libs)
Drag and drop has never worked very well in QtWebAssembly.
But it is working in 6.6.0beta1 single threaded release.
You need to use asyncify though:
Good news is, you no longer have to rebuild Qt to get asyncify, you just need to add a linker argument.
cmake: target_link_options(<target> PUBLIC -sASYNCIFY -Os)
qmake: QMAKE_LFLAGS += -sASYNCIFY -Os