I don't specifically about the Outlander MMCS (yet), but I have experienced numerous lip-sync issues when streaming video from phone to TV, or from my Roku set top box to my sound-bar, or even just watching some streamed content (such as Netflix).
There are so many variables at play, it is hard to work out what causes the issue most of the time. It could be a mismatch in firmware modules between the car unit and the phone. I often find on previous cars that I get weird issues when Apple release a new version of IOS, as my car stereo firmware can't cope with it. It could just be bog-standard latency or some form of split-second internal buffering that Mitsubishi use. It could be a bug in the phone OS, when connecting to certain setups. It could be due to other software running on the phone. It could really be just about anything.
Although it's seldom something I would want to do, it would be interesting to see if folks with other OS's on their phones experience the exact same problem (so iPhone, Windows phone etc).
Does the audio work without lip-sync issues if the audio is sent over USB rather than bluetooth?