Hi.
I finally found the solution. Here it is in case someone comes here by search.
You have to make a POST call:
https://{HOST}/DocuWare/Platform/FileCabinets/{filecabinet_id}/Task/Transfer?storeDialogId={store_dialog_id}
Be sure to add Content-Type header:
Content-Type: application/vnd.docuware.platform.documentstransferinfo+json
HTTP request body:
{"SourceFileCabinetId":"<basket_id>","Documents": [{"Id":<doc_id>,"Fields":null}],"KeepSource":false,"FillIntellix":false,"UseDefaultDialog":false}
If you set UseDefaultDialog to true you do not have to use storeDialogId param.