As I understand the Walk to Read model it simply shifts the location of where the child gets his reading instruction. In other words kids from a 3rd grade classroom walk to a 1st, 2nd, 3rd, 4th or 5th grade classroom depending on someone's estimate of their reading proficiency. This sounds much like the much older Joplin Plan where kids were regrouped across grade levels into more homogeneous classes by reading levels.
No one ever found the Joplin Plan effective so I have no reason to believe that Walk to Read will be an effective plan. Basic problem is twofold. First at least a half and hour every day is lost to we researchers call "transition time." This is measured from when kids stop working in one classroom and prepare to depart for another classroom plus travel time and the time needed to get engaged in the new classroom. All this takes roughly 15 minutes and that 15 minutes is wasted on each end of the period. Or 2.5 hours of instructional time is lost, on average, every week in schools using Walk to Read. Or almost 2 full days of reading periods go down the drain. Only lots of testing could waste more time!
Second, the idea behind Walk to Read seems to be homogenization by reading achievement, meaning that once regrouped the teacher can offer one-size-fits-all reading lesson and more kids will benefit. But when you realize that by 4th grade reading achievement varies from the 1st grade to the 9th grade reading level it becomes clear that even if you were able to precisely estimate reading achievement (and we have no such assessments currently) you would need at least 9 classrooms to reduce the span to a single year span.
What we know is that whether a child learns to read or not the weight of the child's experience belongs on the teacher. In other words, we don't need any more studies demonstrating that 20% of the elementary teachers teach everyone to read and bring low scoring students up to grade level when they are assigned to their classrooms. We also know that another 20% of teachers never develop a full years growth in reading in any of their students. The middle 60% teach the kids we would expect to learn to read but they fail to much advance the reading development of kids who begin the year struggling with reading development and who really need expert and explicit reading lessons.
But the 20% effective teachers are effective with the traditional heterogeneous mix of kids in their classroom. If schools are trying to improve their reading outcomes the evidence is quite clear that most (80%) elementary teachers will need 60 hours of professional development plus one or two years of in-classroom coaching. This will provide the expertise needed to effectively teach all kids to read. Without the PD almost any other initiative will fail. Fail because the vast majority of classroom teachers simply are not expert enough to provide high quality reading lessons. Homogenizing their student pool simply means that most teachers will not effective with a different group of students. The loss of academic learning to transitions will exacerbate the problem.
So to answer your question: What research can be used to resist a Walk to read program? I'd say the research on the power of engaged academic learning time would be the most powerful research to use. Loss of almost 2 days of reading instruction should cause anyone to shudder.
I'd also suggest that no one expects this plan to actually work but, rather, it simplifies teaching and makes a teachers job seemingly easier so in many schools where an administrator is trying to earn points from a faculty this sort of model can be expected.