I very much enjoy the concept that the book is going for, but I feel like it has a few slight issues, particularly with the way that the process of capturing beasts and subsequently tagging them is elaborated upon. The first thing that stands out to me is that there's no specification as to what tags a player can or cannot give to his beasts within the Your Beasts move itself (although the Advanced Moves "Breeder" and "Deeper Training" seems to imply that they immediately apply, say, 1-armor or +n damage to a beast upon capture). Rather, a player (or better yet, a GM) examining the playbook only receives possible examples of tags in the description for the 6-10 move "Mutation", which implies even still that these abilities listed are for a "higher form" of the original beast. I feel that providing a list of example tags off to the side would be, at the very least, useful for sparking the creativity of the player.
The second issue I've encountered is the severity of a 7-9 on some of the moves in this playbook - most notably, "Capture a Beast". I feel like forcing the player to surrender one of their beasts immediately with no alternative is a bit restrictive, particularly with the fictional explanation. The effort drains them - perhaps either they loosen their control over one of their beasts, or they take a debility of their choice? If you still want to retain the singular option, I'd make the fictional effect of the 7-9 a tad more specific ("On a 7-9, you capture it, but you loosen your hold on another spirit to do so ...").
Lastly, examining the Gear section, I'd like to ask a few questions regarding the options provided. Firstly, is something one of the options listed below? And, if so, could you please elaborate on the usage and purpose of taming gear? Moving on, the weapons options for the class seem rather focused towards ranged combat - was this a purposeful choice, and if so, why are there so many ranged offensive options for beasts under "Mutation"? The armors you provide don't make sense either - why does a "buff coat" weigh one point less than a simple long leather coat (ignoring that both outweigh basic leather armor from the DW book for the same level of protection)? If this is all RAW, then this makes the obvious option for any sensible amount of armor the buff coat. And, finally, the miscellaneous gear: why include trap-making gear to start when the move that validates its use is only immediately accessible at 2nd level? What is the fictional function of the map (is it a map to a location where beasts are known to dwell, of the region overall, ...)?
I apologize if this sounds like excessive nitpicking - I truly enjoy the concept the playbook is trying to evoke, but in its current state I feel there are issues with how the content is being explained and integrated into the fiction of the class.