Elliptical image og leather guitar strap end
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Occasional light-coloured clothes staining

(Mainly applies to straps with dyed undersides)

I keep the amount of dye sealant that I use on straps between a minimum and none at all. I don't see the point of using a beautiful, natural, breathing material and then encasing it in plastic.

You will probably come across some of my more-extended rantings on such things in the leatherwork articles.

PLEASE be aware that, if they become damp, any newly-dyed leather product might 'leak' a little dye. Not a lot, generally, but...

I've certainly had shoes that left my feet a bit weird-looking

A white T-shirt and a hot gig with a brand-new strap might not be the best combination.
Yes, I've tried it - well, I had to find out.
It did wash out...

I've never quite established whether it's simply excess dye particles rubbing off the surface, (which would be strange, because most finished articles should have been pretty thoroughly rubbed, burnished and probably waxed), or, more likely, that the dye simply migrates to the surface with the moisture. My only problem with the latter is: why would an oil-based dye respond to water once it has dried? If it does so, why does it seem not to when 'matured'? New shoes usually stain if we are daft enough to get them wet, (even fancy-name ones), but old shoes rarely do. Maybe the frequent slight dampening and salts from perspiration fixes the dye.
Buyer's Advice Clothes Staining Unique Products Why Bother with Leather? Care Guide Straplocks

Leather Treatment and Care