Depending on whom you ask, the reasons for the Bureau's involvement in the Dillinger case might be different.
But regardless of what many think, the reasons are quite clear in a 1933 memorandum found in the file which explicitly shows that while recognizing its limited jurisdiction at the time, the Bureau's assistance in the killing of a sheriff and the escape of Dillinger and others was specifically requested.
You can read the self explanatory details of that memorandum here.
The original Dillinger Squad is shown here which you can see a larger photo of at our photo section in the "navigation" area.