Since Graylog can successfully complete a much more intense search than Kibana, I thought the issue would be related to Kibana. I accidentally closed the browser tab. When I reopened it, two nodes bombed out. I'm allowing the cluster to completely recover right now. Once that's completed, I'll try again and post the errors in the logs.
Please let me know if you think this is still more of an ES issue and I'll re-post in that area.