Equip Retainers with HQ gear

If I equip my miner or botanist with HQ gear does it increase either the quantity or quality of the items retrieved on a venture?

Indirectly it could. As far as I can tell the only stat that matters for DoL retainers is their Gathering score - unlike DoW/DoM retainers where ilvl is what matters.
Higher gathering improves number of items gathered as well as the time needed - up to a limit. I do not know for sure if their Perception matters at all for the quality, or if that is based on their Gathering as well - I think it is all Gathering though.

So, if the HQ gear provides a better Gathering score than NQ gear would, then it would let the retainer improve their haul.

Gathering is definitely key for BTN/MIN/FSH retainers. I've got a couple of retainers with perception gear also equipped and a couple without, there doesn't seem to be a noticeable difference how many HQ items I get when I'm doing the BTN/MIN/FSH ventures.

I wouldn't bother going for HQ unless the item has gathering. And it's helpful to meld cheap Guerdon materia where you can.

Gathering matters for DoL retainers. HQ gear helps when it raises Gathering. It also helps when you upgrade their gear and want to convert their old stuff to materia, too.

I used to think that Perception mattered, too. It does, but it does not. It helps the retainers bring back HQ mats. But really, what good do those HQ mats do anyone in an era where you can HQ anything from all NQ mats anyway? Its not like too many people are willing to pay more for some HQ mats on the board rather than getting the cheaper NQ, they just buy the cheapest on the board.

GP is totally irrelevant for the DoL by the way. So, those accessories are nothing more than a vessel for the highest Gathering materia you can put on them, and then they just sit there never needing upgrading until the next tier of materia gets released.

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Originally Posted by MistakeNot

As far as I can tell the only stat that matters for DoL retainers is their Gathering score - unlike DoW/DoM retainers where ilvl is what matters.

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You have answered a secondary question of mine about my Dragoon Retainer hunting Sea Swallow skins. Based on your answer I boosted my retainer from iLevel 413 to iLevel 450 (all HQ gear). This boosted the overall haul from 7 items to 10 items but made little difference to the number of HQ items. At 413 I was getting 2 out of 7 HQ and at 450 I was getting 2 (or 3) out of 10. Hence, I think iLevel affects HQ very little. On that basis I dropped my retainer to iLevel 431 gear which still provides 10 skins.