Yes and No. If your bar-end shifters are externally routed, then they will not work with the drop grips because your shift cable can not be run between the Drop Grips and the handlebar. But, if you have internally routed bar-end shifting, they can work as long as the cable enters the handlebar at a point away from the Drop Grips.
If you are using the Kitchen Sink Handlebar this will mean you would need to drill holes for cable entry. Here is a video that discussed proper drilling of handlebars. It is primarily related to Di2 wires but can be applied to mechanical shifting as well (2:06 is a fine place to start): https://www.youtube.com/watch?v=cON3w1IL_7k.
Please note that drilling holes in the Kitchen Sink Handlebar can potentially cause unintended issues and will void the warranty and a return or exchange will no longer be possible.