  For a fixed finite group $G$, we study the fields of definition of
geometrically irreducible components of Hurwitz moduli schemes of marked
branched $G$-covers of the projective line. The main focus is on determining
whether components obtained by "gluing" two other components, both defined over
a number field $K$, are also defined over $K$. The article presents a list of
situations in which a positive answer is obtained. As an application, when $G$
is a semi-direct product of symmetric groups or the Mathieu group $M_{23}$,
components defined over $\mathbb{Q}$ of small dimension ($6$ and $4$,
respectively) are shown to exist.
