On the animal side, information is a bit sparer. Outside of mills' own data, which is quite hard to find, SuitSupply provides a social report for the year 2017-2018 which has two pages (out of 53) on the subject.
Though it should be taken with a grain of salt, this report shows wool should not come from mulesed sheep and all animals should be sourced from farmers with good animal husbandry techniques. Leather should be sourced as a byproduct of the meat industry and the animal should be killed before the skin is obtained (yes, it happens...). They are also looking to remove feathers & down from their products which is nice.
I do not know if SuitSupply actually enforces those rules or if they hope outside parties will make sure about it. In any case, this shows the brand is aware of the current fashion industry. Doing it genuinely or as a marketing aspect might not matter that much in the grand scheme of things if humans and animals alike are treated more fairly in the end.
Please do remember that any Code of conduct, sustainability report or other document of the like do not have a juridical value. The brand cannot in any way be held responsible in case child labour, museling or other terrible practices is proven to be a part of the garment's construction.