Clipboard sharing stops working after running on some time on Android 6. Permanent work around: Go to Settings: Battery: Battery Optimization, and select KDE Connect and choose Don’t optimize. Temporary work around: Reopen the KDE Connect app. Based on the work arounds, I guess that the app is prevented from running in the background after some time due to battery policies in Android 6. Sony Xperia Z3 with Sony Concept Software (Android 6.0 preview), latest KDE Connect from the Google Play Store.
This is resolved in Android 7 preview (reconfirmed with latest app version), but leaving it open as many users will be stuck on Android 6 for years to come.
*** This bug has been marked as a duplicate of bug 368438 ***