Konnichiwa!

Due to the necessity of me having to alter the internal clock of my 2DS (to hide my usage of the web browser), I've had the chance to understand exactly how adjusting the internal clock affects time sensitive things in Pokémon Games.

First of all, time sensitive things can be divided into 3 main categories: Things like Berries, that progressively change throughout the course of the day; Daily events, such as daily Battles or handouts; and Special daily events, that are affected more by alterations to the time in the DS/3DS internal clock.

Let me cover Berries first:

Essentially, if you want to make an adjustment to the internal clock, without disrupting Berry growth, all you have to do is pop into the game and save, before you adjust the clock, and then pop in again and save immediately thereafter.

You'll only lose the small amount of Berry growth that occurred in those couple minutes between saves, which means it's fairly easy to adjust the clock without effecting your Berries much.
(The same applies to the various things in Poké Pelago in Gen 7, provided you also pop into Poké Pelago before you save each time.)

Now, daily events are a little different.

Essentially, daily events in Pokémon revolve around midnight, which is when they reset.

If you do the same thing as with the Berries, it'll work just fine.

Additionally, if you actually want to stop the reset of everything that normally occurs at midnight every day (like if you forgot to check the Lottery or something), you can simply adjust the internal clock once without popping into the game first, and it will reset to the day when you had last saved, and all time sensitive things will be as they were then.

Now, things are more complicated for those "special daily events".

Starting with Join Avenue in B2/W2, certain daily events have a special 24 hour lockout that occurs, every time the internal clock is altered, or that you change systems.

In the Gen 6 and later games, all daily events work this way, but in Gen 5, it's just Join Avenue.

Know, it's difficult to circumvent this, but the solution is fairly simple, actually.

All you have to do is make sure you wait 24 hours from when you last saved before you pop in again.

Let me give an example:

Let's say I adjust the date on my 2DS at 1:00, to do some stuff on Paint, and that I log out at 4:00, and turn the date back.

If I popped into the game before I adjusted the date at 1:00, I can do all of the daily events for that day, and as long as I pop in after I adjust the clock back at 4:00, I can do the daily events for the next day, too, provided I wait until after 4:00 tomorrow.
(Due to the 24 hour lockout.)

Were I to pop into to do tomorrow's daily events before the lockout ends, nothing will have reset at midnight like normal, and I will miss out on all of the daily events for that day.

Anyway, that's it.

I suspect a lot of this may be useful to people who play other games affected by the internal clock as well, not just Pokémon players.
(Animal Crossing comes to mind.)

Sayonara!