One thing that could be the problem (besides the old ink supply unit that was emptying carts) is the pump inside of the cleaning assembly.
Basically it looks like this:
If that tubing is worn or the pump is worn, and you are trying to suck air (air from even just the print-head or the flush box->pump area) the pump may not be creating enough of a vacuum to prime the head and get out any air-blocking that may be occurring between the ink-lines and vacuum (there’s a lot of points where air can be blocking between the ink lines and vacuum pump.)
I had this happen to me once in a 4880. My solution was to put distilled water (distilled + simple green, or even better PiezoFlush) into the flush box area and turn the printer on and run a cleaning asap after. This seemed to re-prime the pump/cap motor and I was able to get the air block out.
Generally, this is not an issue on the 3880 though because the cartridges are pressurized. The pump usually does not have to do much work. But if there is air leaking in somewhere, the pump is just sucking air and probably not air from the the head/dampers either.
A test of the pump is to put new colored PiezoFlush or simple green into the flush box and run a cleaning. See if anything reaches the waste ink tank (put white paper towel into the wast ink tank to check). If nothing comes out, than you know that there is something drastically wrong with the pump/cap/cleaning assembly unit. That is not to say that you did something wrong by replacing the ink supply unit. The emptying of the cartridges was a tell-tail sign that the ISU needed replacement. But generally when one thing goes, another is on it’s way. If ink does come out, then you know there is most likely a block above the vacuum area. Either the cleaning assembly is not making contact with the print-head letting air in from the side, there is a total block directly inside of the printhead, or there is still a block between the ink lines and dampers or dampers and head. Or, there is not pressure getting to the carts and your just trying to suck liquid out of closed straws.
There are essentially three components to almost any epson inkjet printer: the ink supply unit (or assembly which includes ink bays, lines, and dampers), the print-head, and the the pump/cap/cleaning assembly. These three moving parts are the parts that die first (or go out of wack first as the case may be.)