(a) has moved
(b) moving
(c) moved
(d) were moving
i think 'moved' is correct
Answers:
Francine and her family ------ to Europe after her mother got transferred three months ago
X (a) has moved (Singular Past) The subject is plural.
X (b) moving (This form ("---ing" is called present continuous. Present Continuous Tense is used to refer to activities happening now. Also, it needs a verb auxiliary such as "are") It is incorrect not only because it does not have an auxiliary verb listed but also because temporally it makes no sense.
(c) moved (Past) You are correct that this is a correct usage. The action of both verbs happened in the past.
Francine and her family MOVED to Europe after her mother got transferred...
(d) were moving This COULD be correct. This is past continuous. This implies that the subjects were in the process of moving when her mother got transferred. To make perfect sense, the sentence would have to continue with some kind of condition, ie, in a conditional sense. For example something like this:
Francine and her family WERE MOVING to Europe after her mother got transferred, but then her mother got a better job in the states.