Do children show top-down processing as much as adults do? There has been much research into how top-down processing works, such as the suggestibility of patients to mishear lyrics based on what the patients are told they are. 

Is there any research to show that this behavior is evident at a young age, specifically less than about 10 years old?
Much evidence suggests children do not show as much top down processing as adults. This depends on your definition so I'll state that I generally mean that use of top-down processing appears to increase as a child develops.

Here is the abstract explaining the results of the study Developmental increase in top-down and bottom-up processing in a phonological task: an effective connectivity, fMRI study.

Emphasis mine:


  In older children, top-down control process may selectively enhance the sensitivity of the LTC to bottom-up information from the FG. This may be evident especially in situations that require selective enhancement of task-relevant versus task-irrelevant information. Altogether these results provide a direct evidence for a developmental increase in top-down control processes in language processing.


Why is this the case? Generally speaking children need to develop a greater understanding of the world around them before they can put Top-Down and Bottom-Up processing to good use. Piaget framed this in terms of Schemas.

Piaget's Stages of Development (which are contraversial mind you) suggest strongly that children start almost at a clean slate and build up schemas as they develop and learn. A younger child simply hasn't had the time to build up schemas for every situation, thus they won't always have a relevant schema to process information in a proper top-down fashion.

Note that other studies suggest that older children have an advantage in some tasks because they learn when to not apply top down processing. Younger children may simply not "know" the relevance of each processing method as well. 

As stated in Visual search in children and adults: Top-down and bottom-up mechanisms:


  The authors argued that with increasing age, children became able to consider multiple dimensions when making relevant comparisons. One interpretation of these data is that the older (but not the younger) children were able to switch off the top-down system to allow salience to emerge via bottom-up processing.