Pokemon-go – Can Pokemon move from their initial spawn location? If they don’t, then what does the lure do

jdero's comment is correct, while it may seem like Pokemon are drawn to the lure, they are definitely spawned on the lure. Areas have a list of Pokemon that are more likely to spawn there so you will see that nearby Pokemon are also the Pokemon that spawn at lures. A few factors make this very apparent. Firstly, in an area with multiple lures activated in close proximity a Pokemon will show up ever 5 minutes or so. The natural spawn rate of any are couldn't support this. Second, I have, while being in one such area, seen naturally spawning Pokemon appear every 20 or so minutes(about average) in addition to the ones spawned by the module. lastly, I have NEVER seen a rarer Pokemon move after using an incense or module near it (within 2 paws), and was able to go out and find said Pokemon after the lure was activated nearby in its original spot. Also incense works the same way as lures except instead of a proximity to spawn in, they spawn directly on top of you.

Eggs are randomly generated. Meaning, there is no correlation between the location where the egg was collected and the Pokemon inside.

Similarly, there is no correlation between the hatch location and the type of the Pokemon collected.

Eggs have been confirmed to be set as a Pokemon type at the time of egg collection. See this resources dump.