If I switcheroo my recruit with an elder dragon and the elder dragon becomes 1 might, kills my recruit from its ability, the dragon is now 1 might on my turn and I play conscription which chooses a 3 might unit and I steal it. Does it work?
Yes, it works.
Here's why, step by step:
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Switcheroo swaps their Might for the turn — Recruit becomes 10, Elder Dragon becomes 1. Elder Dragon's passive makes any amount of its damage lethal to enemy units, so the Recruit (already marked with 1 damage from Elder Dragon's play effect) dies in cleanup because that 1 damage is considered lethal.
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Elder Dragon stays on the battlefield with 1 current Might thanks to Switcheroo's duration.
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Conscription says: "Choose an enemy unit at a battlefield with 3 [S] or less." Per Rule 710, units on the board are evaluated by their current Might — which is 1. That's ≤ 3, so Elder Dragon is a legal target.
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Conscription resolves: you take control, exhaust it, and recall it to your base (it does not go to your hand — recall sends it to your Base). Any gear attached to Elder Dragon stays on it and moves with it under your control (per FAQ #10185).
No hidden complications — the Might check uses the current value, Switcheroo's effect is still active, and Conscription doesn't care about base Might.