Phoenixes are immortal, they die and are reborn from their ashes. But what happens if we blow the ashes as soon as the Phoenix dies? And what happens if we recollect those ashes in a place?
‘Fawkes is a phoenix, Harry. Phoenixes burst into flame when it is time for them to die and are reborn from the ashes. Watch him ...’ Harry looked down in time to see a tiny, wrinkled, new-born bird poke its head out of the ashes. It was quite as ugly as the old one.
Here we seem Fawkes emerges from a pile of ashes, but
Fawkes swooped down in front of Dumbledore, opened his beak wide and swallowed the jet of green light whole: he burst into flame and fell to the floor, small, wrinkled and flightless.
Here we see Fawkes dies and is reborn immediately in mid air, while falling so his pile of ashe is most likely spread out.
Based on this quote it appears nothing would happen should you try to move or separate the ashes, and since the phoenix is reborn immediately you don't really have time to "stop" the rebirth by messing with the ashes anyway.
Side note, phoenix are not immortal,
The phoenix lives to an immense age as it can regenerate, bursting into flames when its body begins to fail and rising again from the ashes as a chick.