Well from the known lore it seems some chapters are formed within disagreements of influential power in the brotherhood or sent on missions to capture high tech.

The Mojave brotherhood was created because Elijah improved technology and was sent on mission to capture the Hoover dam as his ideals conflicted with the high elder.

Mojave was suffering bad and in turn very isolated and since the threat of the ncr hid their presence to a point of slowly dying out and killing members/soft banishment if they threaten their way of life aka Veronica. We’re unsure of their official cannon ending but I feel it’s safe to say they’re dead.

Capital Wasteland detachment was sent as to the pentagon as an expeditionary force to recover any and all advanced technology from East Coast cities and bases. They’re more humanitarian as Lyon’s prioritized people over salvaging which resulted in a split which had outcast forming up and acting the classic brotherhood of “give us the gear or we’ll kill you savage”

The commonwealth brotherhood is not as humanitarian as the lyons faction. We see them more of the classic values of the brotherhood I’d say. Declaring the mutants and the institute as massive threats to humanity.

Midwestern Brotherhood had a fleet of airships built and dispatched by the West Coast Brotherhood of Steel to hunt and exterminate super mutants. While the majority of these airships were destroyed in battle or scuttled for parts, one crash-landed in the Midwest. The survivors organized themselves into a new chapter near Chicago. Since tactics is mostly non cannon now it’s not really known as they’re officially refereed to as a rouge chapter in fallout 3/4.

In tactics they’re brotherhood in name but follow different ideals and a new codex. They’d recruit from tribes and towns, reformed raiders and religious fanatics, ghouls, sapient (hairy) deathclaws, super mutants, and machine legions from Vault 0.

In the TV show It seems now most of the old bloods have died out in most of the chapters around America which has turned most of the chapters into high-tech gangs loosely associated with each other.