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The question says it all.
For example, I want beast boost to boost my speed on Xurkitree. So is it possible or I am stuck with special attack boosts?

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Yes. Beast boost can be manipulated. The way the thing works is by increasing the highest stat after a kill so all you need to do is manipulate the evs, ivs and nature to get ensure the stat you want to get boosted is the highest. For example, you wanna boost Kartana's speed every kill but because of its enormous attack stat, attack is the one that gets boosted. By changing its nature to timid and adjusting the ivs, you can get a Kartana with higher speed than attack thus getting faster with every kill. The same can be applied to Nihilego and while speed boost Kartana is nearly non existent, speed boost Nihilego is the common set used by ou players. Here's how they work

EVs: 252 HP / 4 SpD / 252 Spe
Timid Nature
IVs: 19 Atk

EVs: 80 HP / 176 SpA / 252 Spe
Timid Nature

If you punch this into the calculator, you'll find that Kartana with this set would have 347 atk and 348 speed while Nihilego would have 334 SpA and 335 spe. Upon getting a kill, instead of their regular offensive stats being increaseed, it is instead their speed stat

Having said that, I would imagine that there are limits. If the gap between the ub's highest stat and the one you wanna boost is too high, it might not be possible to adjust the three things I mentioned earlier so that the highest stat's value would be lower than the one you wanna boost

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As this relates to Xurkitree, it is impossible to make its raw SpA lower than its raw Spe (except at level 2 due to rounding quirks).
oh, thank you. I am sad about Xurkitree. I just want to confirm, Fully maxed out speed-boosting nature with 0 ivs in sp. attack and sp. attack lowering nature. It gets tail glow so that is fixed. So anyway? or am I stuck with Z-Rain Dance?
Xurkitree's SpA will always be higher than its Spe after level 7.