Dual of etale fundamental group, field functoriality

Question

Suppose $k$ is an arbitrary field and $X/k$ is a finite-type and integral scheme (hence connected). Let $K$ be the fraction field of $X$. Via the canonical morphism $$\phi: \rm{Spec}(K)\rightarrow X,$$ what is the relationship between the étale cohomology groups $H^1(X, \mathbb{Q}/\mathbb{Z})$ and $H^1(\rm{Spec}(K), \mathbb{Q}/\mathbb{Z})$? Because of the coefficients, these groups are abelian and dual to their respective abelian étale $\pi_1^{ab}$ groups. My intuition says each element from $H^1(X)$ gives an element of $H^1(\rm{Spec}(K))$ via $\phi$, but I'm not very experienced with the functorial properties of $\pi_1$. Are there special cases when there is an injection or surjection between these $H^1$'s? Help is appreciated!