Are lymph nodes that move aren't cancerous?

Not necessarily: We can't tell whether lymph nodes are cancerous or not just by feeling them for movement. The only way to know for sure is to obtain tissue from the lymph node with either a needle biopsy or more definitively, an open biopsy that removes all or part of the node.

Monitoring Centers: Lymphatic system absorbs/carries things too big to go into the arteries/veins like bacteria eaten by white cells and digested food. Lymph nodes "taste" the lymph for anything bad like bacteria/viruses/cancers, etc. When it detects a problem, it alerts the immune system and your body reacts. They enlarge in response to the inflammation they create when they react. They shrink when all is well.