McKay, 23, is a pacy full-back or winger who plays for Crusaders in Super Rugby and provincial side Canterbury.
He will move to Scotland at the end of the New Zealand season, probably in October or November depending on Canterbury's progress in the Mitre Cup, and Glasgow coach Danny Wilson sees him filling the No 15 jersey.
Christchurch-born McKay was part of the New Zealand side that won the World Rugby U20 Championship in 2017, beating Scotland in the group stage. He scored a try in the final as the Baby Blacks thrashed England 64-17.
He made his Super Rugby debut for the Highlanders in May 2018, going on to make 15 appearances for the Dunedin-based side before moving to the Crusaders for the 2021 campaign.
Wilson thinks McKay's pace is ideally suited to Glasgow's artificial pitch.
"We've always known that we needed to add a 15 to the squad and we wanted a real attacking presence," said Wilson. "Josh plays wing and 15, and he's a really exciting runner with the ball, so he'll suit the surface and he'll suit the speed we want to play."
"I like attacking with the ball in hand," McKay told glasgowwarriors.org. "I'm a balanced rugby player and I think I can make an impact on both attack and defence. I really think the style of Glasgow's play will suit me."
Despite his surname, McKay is not Scottish qualified but that is probably to Glasgow's benefit as Wilson has been deprived of his international contingent for most of this season.
"First and foremost, I know that part of my role is developing Scottish players for Scotland - that is the role of the two clubs as well as winning games of rugby," acknowledged the Glasgow coach.
"We take on that responsibility, whether that means re-signing the likes of Scott Cummings and George Horne, or re-signing potential future Scotland players, like Jamie Dobie and Rufus McLean.
"But you do need that quality foreign signing who's here all year round, whether it's in a niche role or he's just a quality rugby player."
The unique challenges of this Covid-affected season have made things much tougher for the Scottish pro clubs, with reduced budgets and an extended autumn international window. But the addition of McKay and Dempsey, two southern hemisphere players in their prime, suggests a significant investment by Warriors.
"Yes, they are big signings for us in terms of where they are in their careers, their age profile," said Wilson.
"They are young men - I am not saying there is anything wrong with signing more experienced professionals to bring leadership, help with the younger boys, and so on, but these guys are about coming in at the peak of their careers."