It is only possible to move items from the Listing which have the same settings as the current Listing ( Attribute Set, Marketplace and Account).
When you click to move items to a different Listing, M2E Pro searches for all Listings which corresponds to the criteria for moving ( Attribute Set, Marketplace and Account). If there are no such Listings, it displays the message: "Listings which meet searching criteria (Attribute Set, Marketplace, Account) were not found.".
You should create M2E Listing with the same settings as a product(s) you want to move ( Attribute Set, Marketplace and Account). See our documentation for more details:
M2E Pro for Magento v1.x:
M2E Pro for Magento v2.x: