I have customers and vendors all over the country, and abroad. We use email, text, phone calls, and video calls to conduct business.I think it's a bad idea to send messages at what you know will be 6 AM their time.
Despite you being busy, you probably should've held onto that email or text message until a couple hours later before hitting "send."
No matter our geographical location, we're all grown up enough to understand that if we receive an email/text in the middle of OUR night, it doesn't require opening at that moment.
If it were a phone call at inconsiderate hours......it should indeed be an emergency, but an email or text will sit patiently until it's recipient takes a notion to open it.
If someone is offended that one took the time to reply/respond with REQUESTED INFORMATION at the "wrong moment" in the recipients day........well then that makes them a whiney vagina.