Finally, after a few hours of power-googling I solved it. Well, someone else solved it for me.
By installing the free app Rungap you are able to import data from Suunto Movescount to Apple Healthkit. It also makes it possible to import data from Runkeeper, even manually entered data. (Currently the Runkeeper app only shows data recorded on IOS8 devices in Apple Healthkit, but by using Rungap everything will show up).
Did I mention Rungap is free? Well, it is. And no ads. If you use it regularly it is possible to make a small in-app purchase to support their efforts. Also, this unlocks some more features, though not needed for syncing in data to Healthkit.