Praise be to Allah, and peace and blessings be upon the Messenger of Allah. To proceed:
The grave is not the final abode of a person; rather, it is one of the stages of the Hereafter. After it come resurrection and judgment, and then the person's final destination will be either Paradise or Hell.
Allah تعالى says:
“Competition for more and more distracts you, until you visit the graves.” [At-Takathur 102:1–2]
Describing the graves as something that is “visited” indicates that a person does not remain there permanently; rather, he stays there until he is resurrected.
Therefore, the expression “He has moved to his final abode”, if it is intended to mean that the grave is the person's ultimate dwelling and that there will be no resurrection or judgment after it, is an incorrect statement. The final abode is actually Paradise or Hell.
However, if the speaker merely means that the grave is the deceased person's final dwelling place in this worldly life, or the last place where he will be buried, then the expression does not necessarily carry that theological meaning. Nevertheless, it is preferable to avoid it and use clear expressions that do not suggest anything contrary to Islamic belief.
And Allah knows best.