Actually, they've supported some gun cases in the recent past.

They, alongside the Second Amendment Foundation, have actually had some success fighting states for discriminatory laws that deny 2nd Amendment rights to specific populations. After all, the Constitution wouldn't work if it doesn't apply to everyone, but states have spent a lot of effort putting up roadblocks to everyone from legal immigrants to medical marijuana patients and states would selectively refuse to renew CCWs and licenses without cause or bar legal gun owners from receiving Section 8 vouchers and imposing categorical bans on legal gun owners attempting to foster or adopt children. In a sense they are the exact opposite of the NRA, who haven't done much to address the lack of equal access to the Constitutional right for quite some time when the complainants don't fit a particular mold.

That's not really true. The NRA has fought for shall issue and even constitutional carry, which removes the discriminatory possibilities under may issue schemes. They worked with an African American in the Chicago handgun ownership case, the PA mother arrested after crossing into NJ, etc. They have shied away from ambiguous or complicated cases, regardless of demographics.