According to Finn, the best method of brushing for mixed dentition:
The four basic methods of tooth brushing are:

Horizontal reciprocating (Scrub method)
Vertical sweeping (Roll and physiologic methods)
Circular (Fones method)
Vibratory (Charters, Bass and Stillman methods)

Key Concept:
Brushing Technique used commonly in children 
The circular brushing method or Fone’s technique is a natural brushing method to use with young children who want to do their own brushing, but do not have the muscle development for techniques that require more co-ordination.
Scrub
Vertical
Fones
Vibratory