The mp4 below shows the last 100 days of ice moving down Baffin Bay (from the Nares and Parry channels) and down the Fram with the east Greenland current. This is a difficult environment for Ascat with so much of the scene either land, open water and passing weather, so the UH AMSR2 fixed land and daily open water masks are used to cut the image down to just ice (though AMSR itself can have weather issues early in the freeze season).
As this transported ice melts out, it brings cold fresh buoyant water to the North Atlantic and Labrador Sea that is a potential concern for the AMOC. However in recent years
none of the ice came from the Beaufort Gyre via TransPolar Drift, not one floe.
In fact this year the Fram ice mostly originated in early November in the Kara and Laptev seas. For that reason, brine exclusion has not preceded very far. Between brine channels within the ice and extruded above to the ice surface, the melt water will have significantly reduced salinity but not be fresh.
As the floes continue to hug the coast until they (or rather their melt) reach the tip of Greenland) the Irminger current has taken matters back up north along the west coast of Greenland and then back south on the other side. We can't monitor this with floe trajectories but in the summer calved bergs from Jakobshavn on reveal directions. Thus there is no support here for the idea that ice from the Arctic Ocean reaches the central North Atlantic overturning zone.
Nares ice is predominantly older thicker MYI but the volume is a tenth or so of the Fram; Parry ice is less still and mostly FYI formed within the CAA but sometimes includes garlic press MYI ice from the Arctic Ocean. Again, Ascat floe trajectories establish that
none of this ice reaching Baffin Bay originated in the Beaufort Gyre during 2017-18, not a single floe. Quite a bit of Baffin Bay ice in the mp4 originates in situ only to melt further south.
Surface ocean currents are effectively the same as surface ice currents according to principal Arctic oceanographer R Woodgate (as ice keels induce them). However most of the water circulates in the Arctic Ocean at a deeper level (~300m) which is not revealed by floe motion nor necessarily parallel to it. Exit volumes along the CAA (125 m western Lancaster), Nares (220 m), Bering Strait (53 m) are somewhat limited by sill heights as only the Fram has a deep sill (2600 m). However these sills primarily impede denser saltier water leaving rather than the more buoyant fresher water nearer the surface.
https://tinyurl.com/y8crdjnh neat poster from 2010 by J Su et al
http://onlinelibrary.wiley.com/doi/10.1002/jgrc.20330/full 2013 CAA flows by C Wekerele et al