Don't think of inflation as super luminal movement of our galaxy from others. Think about the 'stuff' that makes space-time between the galaxies simply increasing (this is inflation). This causes the light to travel through more and more 'stuff' as time is passing by.
Imagine a clump of space-time A, and another B, next to each other at the start of the Big Bang. Then add some more clumps of space-time between them. You wouldn't think that the objects within space-time A and B have actually moved. A really basic analogy would be two pool balls next to each other with an ant on each. The ants can only travel to the next ball by walking on the the balls (ignore the ground). Then imagine the same scenario, but you add another ball between them. The ants haven't moved on the balls, but they would have to travel further now to get to the ball the other ant is on. The balls in the example would represent space-time, and the ants any galaxy/star.
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