The map $\phi_u$ is not always injective. Let $X$ be the blowing up along a large number of points inside a quartic surface in $\mathbb{P}^3$, or any surface $S$ with nonzero $h^{n,0}$. The weight $2$ Hodge structure of $X$ is a direct sum of the weight $2$ Hodge structure of $S$, the image of the pullback morphism, and a finite number of copies of the Hodge structure $\mathbb{Z}(-1)$, the Gysin images of the weight $0$ Hodge structures of the exceptional divisors. So, as you vary the blown up points in $S$, the corresponding variation of Hodge structures is trivial. Thus the Griffiths transversality map $\phi_u$ is zero for every $u$ coming from a variation of the points.