@Vesselin and Jason: thanks for your comments, but I am not sure if these degenerations that you mention (e.g. double of Kummer) can arise in my situation, since the total space is smooth, and the fibration is the Iitaka fibration, which gives restrictions in general. For example, when $X$ is a surface, if the Iitaka fibration is given by sections of $|K_X|$ ($\ell=1$) then there are no multiple fibers at all. However, I feel that using Grothendieck's Stable Reduction for abelian varieties should be helpful, but I couldn't exactly see how... maybe you know how to do this?