What causes the print heat to 'catch' on the part? What would cause this effect, and how can it be avoided?  

This is a PLA print, and it should look like this. I tried 215 and 225°C and both had the same effect. An earlier similar print at 220°C was not as bad but it still had some catching- it seems hit and miss and not strongly related to extrusion temperature. 

Outside: 



Inside: 



I'm using a QIDI dual extruder printer with Makerware 2.2.2.89 software and these parameters: 





Speed is 60/80mm/s.
I think that you should first verify that you have the latest firmware and a newer version of MakerWare. I experienced similar issues around that version of MakerWare and remember a lot of print errors came with it.

I believe the MakerBot Dual firmware is somewhere around 7.? and is no longer in development.

MakerWare is also no longer in development as a standalone application and seems to have been ported over to the MakerBot Desktop. However I've personally found v2.4.1 to be substantially more stable than v2.2.

I have not tested it, but supposedly the new MakerBot Desktop (v3.10) is compatible with the Replicator Dual "Original". I had tried an earlier version of MakerBot Desktop and reverted back to MakerWare 2.4 because I ran into issues with connectivity. However, I'm not certain it was an issue with software so much as the exploding voltage regulator...

It might be best to give the new software a shot and/or try v2.4 of MakerWare, if you can find it.

Here's the link to the latest MakerBot Desktop

Here's the release notes for MakerWare/MakerBot Desktop v1.0-latest (v3.10+)