I searched for answers on the Apple Support forums, and, as usual, that is the worst place to go for help.
Here is what I have dug up and I posted over there:
Looking around the Internet, this is a HUGE issue with El Capitan (EC). Home Sharing is not working correctly between desktops running EC and Apple TV (4th Gen). There are a dozen or so different work-arounds that people have figured out, but none of them seem to stick. Every day I look for a new update to either iTunes, Apple TV, or EC that addresses this issue, but no luck thus far.
Fix #1
1 Open System Preferences from the Apple () menu.
2 Open Sharing pane.
3 Double-click the name of your Mac in the Computer Name field
4 Delete and rename your computer
5 Close Sharing pane. System Preferences will automatically saves the new name.
6 Go to iTunes.
7 Quit iTunes and launch it again
Fix #2
1 Open iTunes
2 Open preferences
3 Click Sharing icon
4 Ensure you have checked the box for sharing your library on your local network
5 Quit iTunes and launch it again
Both of these "fixes" work for some people and not others, and most of the time are intermittent - sometimes movies will cut out while people are watching them. This is a HUGE issue that I wish Apple would address. I can't stream movies that are on my desktop's iTunes!!