Pre-Cache previous and next months' events?

I’ve got loadMode set to month.

I’d like to also preload the next month and previous months data. If the user moves to the next month, I’d like to also pre-load the month following that, and so on.

Does the scheduler have any support for that?

What would be the best way to add that functionality?

There is no such built in functionality.

In theory it possible to define custom time units (something like “nextPrevMonth”) and use it in setLoadMode.

You need to define the next methods
scheduler.date.nextPrevMonth_start = function(date){}
and
scheduler.date.add_nextPrevMonth = function(date, inc){ }

That would be one way to do it, but what I’d really like is to load the next and previous months in the background while after the current month has already loaded.

Can you think of an approach to do that?

The above solution is the only way to affect the data caching logic.
And if correctly implemented - it will work as necessary in your case, for each data request it will preload prev and next month.