At my last job I managed a team of developers in India (while residing in the US). It was pretty much necessary for me to be available outside of my company's normal work hours. I always compensated myself for middle-of-the-night activity with time off during the day and nobody ever mentioned having a problem with it. I was eventually rewarded by being laid off with everybody else when my company was acquired by a west coast tech giant.
To prep my day. As a late shift I want to know what I am walking into rather than be anxiety ridden for my 4 hours of day light. That being said I don't respond I just check to see what is happening