The implication is not true in general, if I understand you correctly. For example, consider the property "affine" for morphism of schemes. 
Say, you have a morphism of schemes $f:X \to Y$ such that for every artinian ring $A$ and morphism $\text{Spec}(A) \to Y$ the base change $f_A : X_A = X \times_Y \text{ Spec}(A) \to \text{Spec}(A)$ is affine. Then $f$ is not an affine morphism, in general. For instance, consider for $f$ the open immersion of the pointed affine plane 
$\mathbb{A}^2_k -\lbrace0\rbrace\to \mathbb{A}^2_k$.