Property of a morphism inherited by the fibers Let $T$ be a scheme over an algebraically closed field $k$ and let $f:X\to Y$ be a $k$-morphism over $T$. 
For which properties P is it true that if $f_t:X_t\to Y_t$ has P for all geometric points $t\in T(k)$, then $f$ has P? Is it true for open immersions, for instance? What about properness, flatness...? (Feel free to add some assumptions, I really have no idea whether I am asking something very strong or not!)

No, open immersion of fibers does not imply actual open immersion.  
For example take $T=\mathbb A^1_k, X=\mathbb A^1_k, Y=\mathbb A^2_k$
For the structural morphisms  $X\to T$ take the constant map  $x\mapsto 0$ and for $Y\to T$ take the first projection.
Finally for $X\to Y$ take $x\mapsto (x,0)$, which is not an open embeding but is a fiberwise open embedding.
