Red Shift Question

[Fly135]

I've been confused about how red shift works.  If I understand correctly when we look far out into the cosmos stars that are farther away are more red shifted.  My question is... Does the red shift predominately come from the stars speed moving away from us, or does the light also red shift as  it moves through an expanding universe? I.E. a combination of both factors.

Given that it's said the expansion of the universe is accelerating, you would think that the further away you look (i.e. father back in time) that the red shift would be less.  That is if the red shift entirely occurs when the light leaves the star.  If the red shift also occurs as the light travels to us then I could understand that more red shift occurs the farther it travels.

[StringJunky]

The stars photons get more red-shifted as they travel through the expanding space. The longer they travel, the more RS they become.

[Fly135]

23 hours ago, StringJunky said:

The stars photons get more red-shifted as they travel through the expanding space. The longer they travel, the more RS they become.

Thank you for the answer.