So yes, you can initiate a grapple on a successful bite action (claws also if in Gauru). If your victim is later in the initiative, they basically lose their turn. When it comes back around to the higher of the initiative of the grappling characters, the first contested Strength+Brawl vs Strength+Brawl roll is made to figure out which gets to act.

Now, this actually reminded me of a question I had:
If a werewolf in a non-human form wins a grapple contest and chooses to do damage, that damage is lethal instead of bashing, right? (Unless they want to do bashing in Dalu.)
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