Im wondering if this is your body's way of showing an emotion about the conversation that you dont realize you are showing. reason I wonder this is because a mouth turning down and feeling like bursting into tears is the normal human way of showing the emotion of sadness, displeasure and sometimes anger.

suggestion next time it happens maybe you can look back on the conversation to see if the conversation was causing you to feel sad, or any other feelings that may have made your body physically react but you may not have been emotionally connected to.

I sometimes have this problem with both my depression disorder and my bipolar disorder. but I also have this problem with my physical problem called MS (Multiple Sclerosis)

facial tics besides mental and physical can also come from medication side effects \problems..

my suggestion would be to see your medical doctor or any other treatment providers you may have. they will be able to tell you why you are having this problem of your mouth drawing down ward and feeling like you want to burst into tears at the same time.