A new version of Pushbullet is rolling out today that allows your clipboard to transcend the constraints of a single device. Just flip the switch, and all your devices will automatically sync clipboard content through Pushbullet. Cool, right?
You'll find the toggle for this feature in the advanced settings. The default setting is "off," so you'll need to activate it on every device with which you want to use it. While the intention is for the clipboard sync to be actually universal, it's currently limited to Android devices and Windows PCs via the desktop app. The browser extensions aren't capable of this functionality. The devs are working to implement clipboard syncing in iOS and OSX, if you have any of those devices sitting around.
The update is live in the Play Store, and it does seem to work as advertised. Clipboard content is passing between my devices in just a few seconds. As always, Pushbullet is free.