1. When we depended on the Postal Service and the telephone, we knew that letters did not demand replies in the next few days, perhaps unless they were sent Special Delivery. More urgent communications depended on the telephone and telegraph.
With email, we might feel the need to reply promptly--in part to clear your Inbox! But also, because that has become the expectation. But, in general, you might do well to wait until you have time, except for confirmation of meeting times. Assume the message is just another Postal Service letter. If you have the discipline (I do not), look at your email once each day, much as you might your mailbox.
Moreover, much mail that would seem to demand a detailed response is best left unanswered. Long complex complaints might well be dealt with in person. Whatever you write will engender another demanding reply. Even if the letter accuses you of heinous crimes, and has been cc'd to your boss or colleagues or..., let it go, maybe with a brief response to the boss. ["I am not sure what Joe is talking about." is likely to be enough, since the boss has real problems to deal with.] Of course, if there is any substance to the accusation see an attorney, and surely do not respond at all.
Much the same for accusatory emails from colleagues, significant others, etc. Let it go. I have inadvertently deleted my Inbox and my Deleted Mail, so I even have an excuse. A telephone call or a meeting may be in order, but email escalates too rapidly.
2. Of course, a meeting or a telephone call can become an Inquisition. If that is what is going on, just drop the call--it happens all the time in any case. If they call back, you have already gone to the restroom. As for a meeting, in the middle of an in-person Inquisition you can always plead a medical appointment, or if you are so inclined, projectile vomiting with collateral damage on your accuser is quite powerful.
3. Always be sympathetic, always be polite, always say they can send the message again, always suggest you reschedule, and call them back in two weeks to see if there is something they had in mind not said already. And always keep a copy of the emails from the complainers.
Of course, if you are involved in a conspiracy, as in much of Wall Street, no email, no phone calls. In person, and ideally checking for the presence of recording devices. And if you are copying, with the potential for plagiarism, check it out on Google or Turnitin to be sure the your copied passages are not recognized there--perhaps someone else has copied them too.