Gmail API Puzzle- stop emails from moving

Some of my emails keep disappearing from my inbox. Most of the time it is when I get several emails with the same subject (the body is different). I will label one of the emails and then 5 of them will move out of my inbox. I can find them when I look for all emails. I contacted Google, they said an API is moving the emails but they cannot tell me which company is doing it. Can someone tell me what company is moving an email? or can you get it so it stops moving out of my inbox??