Time to wash two identical down jackets for the first time.

Care instructions are straightforward except for one item: label says to use a mild powdered detergent.

Mild I get, sure, but it's an HE washer and my current detergent stash is all liquid. Looking around at HE friendly powdered detergents, I don't think any are actually "mild", despite being labeled as "gentle". When I think "mild" detergent, I think Woolite or similar, not a gussied up Tide that is just scent free.

So two questions: is it actually OK to use a liquid detergent on these coats, despite the care instructions? If it's really not, what is an actually mild, powdered, HE-friendly detergent?

I've used liquid detergent just a small amount with the washer on a gentle cycle.

I've also used Tech Wash products before but for adding some water repellant back to older jackets.

https://www.nikwax.com/en-gb/products/

I always use Nikwax down wash for our down jackets.

I am used hair shampoo to get the oily places on the collar before washing, Just pour some liquid on the oily places and rub it in before washing.

Babu shampoo....$ store. Wash on Delicates

Wrong time of year to wash your down, having a hot sun outside to help the drying process is big.

You don't need much detergent, in fact, it's best to do a double-rinse minimum. You absolutely need to make sure the detergent is off the down clusters or, they'll dry-up prematurely and/or clump because of the soap residue left behind. When in doubt, rinse.

Make sure your washer is a front-load, DO NOT us a top-load washer. Your jacket will get wrapped mourned the center agitator and will result in tearing and over-stressing of the stiching and baffles. Some people will only do washings in their bathtubs. I've always done my down washing in the tub, didn't wanna chance it with a machine. Drying is just as important as the amount of detergent used. Seek out the largest drier you can find, you want your garment to tumble around; Putting in 2-3 sneakers so that they continually break-up the wet down that wants to clump together. Some people like to use tennis balls, unless you're putting in a lot, I find some clean sneakers more effective.

Some advice from leading down purveyors:
Western Mountaineering
Feathered Friends

Hand wash in the tub.