And rarely have I met a graduate of a general clinical track that compares (in terms of expertise, previous experience, mastery of the full range of child related assessments/interventions) as those that graduate from child clinical programs such as that offered at Kansas.
Fresnel, you're meeting me, soon-to-be graduate of clinical program that offers lifespan developmental training.
I'm leaning more towards LisaLisa86's opinion. As she said, what if you were asked to work with families where the 'identified patient' is a child, parent-child dyads, or late adolescents/emerging adults (developmental stage just beyond adolescence when these folks are not yet mature adults w/ 'adult presenting problems')?
Another option is to go into combined clinical/school psych programs, like Hofstra in NY (not where I attend though, but it's a great program). You get a ton of exposure to child assessment in these programs.
If you're starting off, entering any program that gives you lifespan training is best IMO. In my last years of this type-program, I focused on child work through coursework and externship placements, and I only applied to a few child track internships. (I could have opted for exclusive child externships, but I branched out for exposure.) But I will work with children in the future b/c it's an art (in addition to the skill) and I'm good at it. Sure, there are others with exclusive child-focused training and dissertation topics on children, but in no way would that type training make them better, more effective "child clinicians" than me (when in the actual therapy room w/ a child). My theoretical, intervention and assessment experience has trained me with a solid base from which I can "specialize" by post-docs, job placements, etc.
By the way, the confidence in my statements comes from having been through lifespan training, having clinical experience with traumatized children, and being a mom of four (the latter mainly gives me prespective, but I'm not in any way implying that it is a prerequiste and definitely NOT suggesting others go out and have a bunch of kids to find out prior to graduate school). But, I'll tell you having worked with severly mentally ill, psychotic adults and learning their personal histories gives me a ton of insight into what happens when interventions are not implemented in early childhood (and the significant roles of mother-infant attachment and family dynamics). I could actually be so bold as to suggest looking into some psychodynamic-focused programs because you will get all this training and more.
It is important for you to focus on your interests once in your accepted program, not necessarily submersing yourself in a clinical program that focuses exclusively on children, unless you want to do school psychology (which is often one of the main areas that problems are detected).
Apply everywhere (avoiding those programs that offer minimal coursework in child development, child psychopathology, child psychotherapy, child assessment) and keep your focus. Good luck!