Does childhood obesity persist into adulthood?
 

Obesity during infancy is not a strong predictor of obesity in adulthood but the risk increases with increasing age of the child, degree of obesity and the presence of parental obesity.  Obese children and especially obese adolescents, are more likely than normal weight peers to become obese adults but only about one third of adult obesity develops out of obesity manifest already in childhood.

Time course of fatness changes in childhood- the adiposity rebound concept
The BMI increases after birth until about 6-12 months of age, then decreases to a minimum in early childhood (usually at about 4-6 years).  Thereafter  the BMI curves starts increasing again and steadily increases until it reaches a plateau in young adulthood.

The age at which BMI starts rising again is known as the adiposity rebound age, and has shown a close association to the risk of obesity in adulthood. It is considered to be the best predictor during the preschool age of later obesity risk in young adulthood. The earlier the adiposity rebound, the greater the likelihood of adult obesity.