How to determine polarity of components in TLC
I am assuming this is TLC run on silica. Silica is a mildly acidic surface so basic compounds such as amines do not come off the baseline unless you use a very polar eluent with added base (e.g. ammonia/triethylamine). Starting material B is a basic amine. Starting material A is an epoxide and fairly non-polar (kinda similar to an ether) and will run high in most systems, certainly any system where B comes off the baseline. Your product C is more basic (tertiary v secondary), this is the strongest effect. The second non-polar aromatic group raises the likelihood that it will run above B, but this is a weak effect. The alcohol is a polar group and will also pull it down thus the sum of these effects is that it runs lower 

This missing information is the eluent. From what what is given in the question it is clear that the eluent is something like DCM/MeOH/NH3 (that would be my first attempt to separate such basic components). As B is in excess what you are really looking for is loss of A. When you can no longer see any A by TLC, the reaction can be worked up.

The groups that make for low running are those that interact strongly with the silica surface: basic amines, acids, alcohols. This is a general rule of thumb, but in my experience trying to guess where a multi-functioned molecule is going to run is quite a lottery. I do not believe that it is a safe assumption that a product similar to C would always run below a SM similar to B.