I'm quotating from dictionaries.
【大辞林】
動詞の連用形またはこれに「て(で)」の付いた形に付いて、動作が進行し、また、事態が推移する意を表す。
【広辞苑】
(8)(動詞の連用形またはそれに「て」を添えた形に付いて)
(ア)次第にそうなる、また、そういう状態が出現する意を表す。
It is really difficult even for the natives (or because I'm native?) to explain exact different between
人によって違う and 違ってくる
先生によって変わる and 変わってくる
My mental image is 違う 変わる just state the fact as it is something already exist, while 違ってくる 変わってくる shows the situation has not been fixed yet, and if it takes this turn, it becomes (kind of change) to this, or that turn brings that... as if things and evens are in progress in front of you.
This sense of movement has something to do with... with... with.... sorry I cannot put it into words
Ohhhh.....
急に暖かくなって随分と夏らしくなって きました。
When I hear this sentence, I feel the the change (becoming more summer like) is rather a gradual change, even it is caused by the "sudden change of temperature" which itself is not something gradual.
Maybe my explanation is not very clear or understandable, because I'm talking based on just my kind of instinct/feeling of a native, it may not sound very logical.
I hope my poor attempt to convey the feeling behind the expression or connotation or whatever, gives some ideas about てくる.