Does new Bami's Cinder passive work with slows, more specifically gunblade slow?

no

Nope. The change mostly effected Leona. Shes a A tier jungler now. Cinderhulk proc+Sheen/Gauntlet/Trinity(if ur rich) proc = ouch, whilst being tanky.

It also helped champions they weren't meaning to buff like Leona and, my favorite, Lissandra. Now you can build Lissandra with aftershock and fully use it, also build her bruiser style and still do damage with Sunfire, Abyssal, and Liandry's as your core.