  We extend a technique, originally due to the first author and Poonen, for
proving cases of the Strong Uniform Boundedness Principle (SUBP) in algebraic
dynamics over function fields of positive characteristic. The original method
applied to unicritical polynomials for which the characteristic does not divide
the degree. We show that many new 1-parameter families of polynomials satisfy
the SUBP, including the family of all quadratic polynomials in even
characteristic. We also give a new family of non-polynomial, non-Latt\`es
rational functions that satisfies the SUBP.
