Nothing is instant.
Instant = 0.0000 sec. valid or invalid
"Blip of a eye" maybe is.
FTG need to mark the location.
->Either by special kunai, that was mark (seal); or focus the target, which Minato did.
Kamui is about dimension; of which belong to the rightful owner;
Obito did said about that he required a lot of chakra to sync at Kaguya dimension.
So, it also mean that Kakashi required a lot of chakra in order to kamui, as the eye belong to Obito.
Now that Kakashi without that sharingan eye, his chakra return to normal with reasonable normal ninja chakra.
Kamui is not instant.
If it is instant, how can Obito focus to sync and kamui to Kaguya dimension.
Manga fact.