The answer depends on the version of the Chern classes you are using.
For example, if you consider Chern classes with values in the Chow ring then the answer is negative. The simplest example is the following, take $X = T$ to be an elliptic curve, and $F$ to be the line bundle corresponding to the diagonal. Then $c_1(F_r)$ is the point $r$ itself, but different points are not rationally equivalent.

On the other hand, if you consider Chern classes with values in cohomology with rational coefficients then the answer is positive (if $T$ is connected!). Indeed, since $X$ is smooth and projective any flat sheaf has a resolution by vector bundles. Because of the flatness its restriction gives a resolution of the restricted sheaf. Since Chern classes are multiplicative in exact seqeunces, the question reduces to the case of a vector bundle. Further, because of the splitting principle we can assume we can assume that $F$ is a line bundle. And in this case we can deduce everything from Riemann--Roch.

Note by the way that the same argument shows that Chern classes are also constant in non-flat families, if you only replace the restriction $F_r$ by derived restriction.