Step 1 -- rule out dates/times that are not possible
Start by sending an electronic-mail message asking for each person to summarise his/her availability during the relevant few months. This may not immediately elicit an obvious date or time, but allows you to rule out dates and times which are definitely not possible, thus sparing everybody the hassle of keeping lots of specific options available.
Step 2 -- suggest some specific options
Compile a set of numbered/lettered options, again by electronic mail. A message might read something like:
Dear all,
Further to our previous correspondence, it looks like one of the following options is likely to be best:
a) 29th February 2100 in the morning;
b) 29th February 2100 in the afternoon, finishing not later than 4pm (NB: Prof. X has a train to catch at 5.20pm);
c) 31st April 2100 at 11am (NB: Dr Y is giving a lecture in the same building at 9.30am–10.30am);
[a few more options]
The numbering/lettering of options is vital, since this makes life easier for writing quick replies, such as:
Dear all,
a) most convenient for me
b) possible (but would need hotel that night as well as previous night)
c) not sure, since it is around the time of the International Conference of Futile Studies, for which I am the keynote speaker
[comments on other options]
Yours sincerely,
external examiner
Or even...
Good -- a, d, e
Possible, but not ideal -- b
Impossible -- c
Crucially, numbered/lettered options are amenable to quick updates such as:
Dear all,
UPDATE: sorry, option c) not possible (unless the university can charter a private helicopter to take me to the conference immediately afterwards!?!)