
Leads usually get used so much in our generally clean (cleaner than a horse!) hands, and dogs don't sweat like horses do, so doggy stuff does tend to stay cleaner and more supple than horse tack will.
However if you get things wet and dry them, especially if they dry fast in a centrally heated house, leather will go hard and nasty, so slow drying and then massaging in a very TINY amount of oil (too much is baaaaaaad) iwth y our nice warm hands will do the trick.
If leather gets properly dirty, especially greasy dirty (inside of collars for example), warm water wtih a tiny dash of washing up liquid to clean it with and then wax/fat based conditioner afterwards again massaged in with your warm hands - its the warmth of your hands and the rubbing it in that makes the leather soft and supple, NOT the amount of product you use!
Saddle soap - the glycerin based traditional stuff is frequently mis-used - it is a humectant (draws moisture in) which is why it keeps leather supple, but if its not removed properly and the item is left alone it will in the end go mouldy!