Caitlyn & Thresh Synergy
Caitlyn ADC and Thresh Support win 51.0% of 20,250 Low Elo games together, +0.2 points vs expected, 2-patch window 26.19-26.20.
Researched on LoL patch 26.20. Tips are written to hold at any rank; stats come from Low Elo ranked games up to patch 26.20.
Playing the duo
Death Sentence(Q) stuns and briefly pulls the target toward Thresh; its recast moves Thresh toward the target. Caitlyn can place Yordle Snap Trap(W) during the stun, and a sprung trap grants an empowered Headshot(P). Flay(E) can push an enemy onto a trap if it is positioned well.
Caitlyn's only mobility is the recoil of 90 Caliber Net, so Thresh's Dark Passage lantern - which she can click to dash to him - is her main escape from ganks and dives, while its shield soaks damage during trades.
How to play as Caitlyn
When Death Sentence(Q) lands, place a trap under the target during the stun if Caitlyn can safely step up; Thresh's recast moves him to the target, not the target toward him again.
How to play as Thresh
Hold Dark Passage for Caitlyn when the enemy jungler dives or she gets engaged on, because apart from her net recoil she cannot escape; a lantern out of range of the divers is worth more than a shield used in a trade.
Use The Box(R)'s slow to make a trap setup or a Headshot(P) that is already primed easier to land; the slow itself does not empower Caitlyn's attack.
Sources
These notes come from LoL Wiki and r/ADCMains, checked against the champions' abilities.