  Given a $G$-Galois branched cover of the projective line over a number field
$K$, we study whether there exists a closed point of $\mathbb{P}^1_K$ with a
connected fiber such that the $G$-Galois field extension induced by
specialization "arithmetically descends" to $\mathbb{Q}$ (i.e., there exists a
$G$-Galois field extension of $\mathbb{Q}$ whose compositum with the residue
field of the point is equal to the specialization). We prove that the answer is
frequently positive (whenever $G$ is regularly realizable over $\mathbb{Q}$) if
one first allows a base change to a finite extension of $K$. If one does not
allow base change, we prove that the answer is positive when $G$ is cyclic.
Furthermore, we provide an explicit example of a Galois branched cover of
$\mathbb{P}^1_K$ with no $K$-rational points of arithmetic descent.
