In the old (before version 4) LinkedIn application, there was a button to Download all connections to my iPhone.
In the latest (4.0.2) I cannot find that option.
How can I update (download) all my connections with the latest LinkedIn version?
Considering that their mobile app page doesn't show the current version, and the current version doesn't have this feature anywhere that I could find, I'm going to say that they have removed this feature.
Try using their feedback page to request this feature in v4.x, or an explanation as to why it was removed etc..