‘ And the children of Israel gave to the Levites out of their inheritance, in accordance with the commandment of YHWH, these cities with their suburbs.'
The response of the people was immediate, and cities were allocated by lot to the Levites. There was no guarantee that those cities were all available to dwell in. Like the general allocations they had in many cases still to be possessed. It was an act of faith, just like the allocation of the land, that God would give them their inheritance.
The Levites are regularly described as ‘sojourning' in the land (Deuteronomy 18:6; Judges 17:7-8; Judges 19:1). They were not to have permanent possession like the other tribes for their inheritance was YHWH (Numbers 18:20; 23:62). They had to have places to live in throughout the tribes so that they could fulfil their function, but these were not to be seen as their own but as lent by God. This was the ideal. And at the great feasts theirs was the responsibility of ministering at the Tabernacle (Numbers 18:22 compare Deuteronomy 18:7), although the priesthood itself was restricted to the ‘family' of Aaron (for example, as well as in Leviticus 8 and regularly in Numbers, priests and Levites were clearly represented as distinct from each other in Deuteronomy 18:3-8). It is quite probable that suitably dedicated people could be adopted into the tribe, and indeed into the family of Aaron, as considered appropriate (e.g. Samuel), just as they could be adopted into any of the tribes, but all was no doubt done ‘before YHWH'.
Allocated along with the cities provided for the Levites was to be a certain amount of land for their cattle (Numbers 35:4-5), an ideal probably never fully realised. The distances were ideal representations and not to be treated literally. The idea would seem to be that the city was to be seen as within a square each side of which was 2000 cubits, similar to idea of the square of the Holy of Holies. Thus the first 1000 cubits of land around the city belonged to the Levites. They indicated the ‘holiness' of the cities as being dwelt in by the representatives of YHWH (compare Joshua 3:4). But this land could never be sold (Leviticus 25:34).
What precisely was meant by the giving of the cities is not absolutely clear. They were certainly not given the cities outright with the inhabitants moving out to make way for them. What was probably given to them was a portion of the city, or selected houses within the city (see Leviticus 25:32-33 which only makes sense if Levite houses were in general walled cities), and land close to the city walls. (Certainly Shechem did not become a totally levitical city - Judges 9. Nor could that idea have been in mind in the short, or even middle, term in view of its nature). It is possible that there was in mind in the distant future that eventually most would become wholly levitical cities.
That the cities were spread throughout the land was necessary because the Levites represented the redeemed firstborn of the children of Israel (Exodus 13:2; Exodus 13:13 with Numbers 3:40-44) as separated to the service of YHWH.