They are two friends (a sparrow and a woodpecker) who are seen either helping Big Mama act out a plan or trying to catch Squeaks, a caterpillar. Dinky and Boomer are seen at many times trying (unsuccessfully) to catch him. At the end of the film, Squeaks becomes a butterfly and flies away.
Dinky the sparrow and Boomer the woodpecker act as the comic relief "odd pair" during the film, Dinky as the small but brainy half and Boomer as the strong but dimmer half. They help Big Mama find a home for Tod by getting Widow Tweed's attention (Boomer taps at her door and Dinky helps upset her clothesline).
As Tod starts growing up, he longs for a playmate, but the bird duo is too busy trying to catch the caterpillar Squeaks to notice. Next, while Amos Slade and Chief were chasing Tod on his property, the bird duo were nearly shot and quickly flew away. Next, after Copper leaves at the start of hunting season, the two help Big Mama illustrate her point that things are most likely going to be different between Tod and Copper, due to the later being a hunting dog. They fly south for the winter and return in the spring, amazed and proud of how much Tod has grown up.
After the accident of that night, when Tod's safety is in jeopardy, Widow Tweed takes Tod to a nature preserve, and Dinky and Boomer accompany Big Mama to watch his attempts at making friends with Vixey. At the end of the film, their attempts to catch Squeaks are as determined as ever but are dumbfoundedly foiled when Squeaks becomes a butterfly.