How do brush battlefield tokens work? Do they ever disappear or do they stay all game?

A Brush battlefield token is a domainless battlefield token that provides a passive buff and has a specific replacement ability. According to Rule 184.8, its text is:

• "Bird, Cat, Dog, Poro, and Ivern units here have +1 [M]"
• "When you score here, you may replace this with the battlefield it replaced."

Regarding your question about whether they disappear: They do not disappear automatically. They remain in play as the battlefield until one of two things happens:

1. You choose to replace it: As stated in the token's text, when you score at that battlefield, you have the option to replace the Brush token with the original battlefield card that it replaced when it was created.
2. Game effects: Like any other battlefield, it remains in play unless a specific card effect or game rule causes it to be removed or replaced.

In summary, the Brush token stays in play until you choose to swap it back for the original battlefield upon scoring, or if another game effect removes it.