I've never personally encountered this, but from the way the deeplink module works I can tell you what's going on and why this happens. The short version (for when you don't want to read this entire wall of text): The deeplink module assumes only one link will be opened at the same time per user. When you open multiple links at the same time, the links that are executed first can get overwritten by the later ones before they are completely finished. This causes the behaviour you see, where it functions correctly when debugging (slowly), but not normally.
Unfortunately, there is no easy fix using the deeplink module as it currently is, other than slowing down the rate at which links are rquested. I suggest you file a support request to make this possible in the deeplink module. If possible provide a test project, and reference this forum post to explain the issue.
Now for the full explanation:
The deeplink module handles a deeplink request roughly as follows (modified from comments in the deeplink java code):
As you can see, the module removes any "outdated" pending deeplinks when a new link is received. This is done because the module assumes only one deeplink will be opened at the same time for a single user. In the most common cases (e.g. opening a link from an e-mail, or linking to a profile page) this is correct.
In your case, you are trying to open multiple links at the same time. When done slowly (with debugging on), this is equivalent to opening a number of links quickly, but sequentially. If it's done too fast, the java action for the next link can run before the previous link has loaded completely (into step 8).
This causes the previous pending link to be deleted (step 5). When that link gets to step 8, it loads the pending link created for the next link, because that is the only one present for the user. Then the pending link is removed (step 9), so when the next link gets to step 8, it has no pending link anymore and will display the default home screen instead.
I hope this sheds some light on the issue.