There are only a limited number of studies on this, with a limited sample size. None of them seem to check if children "changed their gender", but if they still match the criteria for GID.
Here is one looking at 25 girls, only 15 of which actually met the criteria for GID. At follow-up, 3 met the criteria (12% of the 25 girls total, some of which did not meet the criteria at the beginning either):
This study provided information on the natural histories of 25 girls with gender identity disorder (GID). Standardized assessment data in childhood (mean age, 8.88 years; range, 3-12 years) and at follow-up (mean age, 23.24 years; range, 15-36 years) were used to evaluate gender identity and sexual orientation. At the assessment in childhood, 60% of the girls met the Diagnostic and Statistical Manual of Mental Disorders criteria for GID, and 40% were subthreshold for the diagnosis. At follow-up, 3 participants (12%) were judged to have GID or gender dysphoria. Regarding sexual orientation, 8 participants (32%) were classified as bisexual/homosexual in fantasy, and 6 (24%) were classified as bisexual/homosexual in behavior. The remaining participants were classified as either heterosexual or asexual. The rates of GID persistence and bisexual/homosexual sexual orientation were substantially higher than base rates in the general female population derived from epidemiological or survey studies. There was some evidence of a "dosage" effect, with girls who were more cross-sex typed in their childhood behavior more likely to be gender dysphoric at follow-up and more likely to have been classified as bisexual/homosexual in behavior (but not in fantasy). A follow-up study of girls with gender identity disorder.
Another study looked at 59 boys and 18 girls. 30% did not respond to a follow-up, and 12 boys and 9 girls still met the criteria for GID at follow-up (20% and 50% respectively, or 30% and 64% excluding those that did not follow up):
At follow-up, 30% of the 77 participants (19 boys and 4 girls) did not respond to our recruiting letter or were not traceable; 27% (12 boys and 9 girls) were still gender dysphoric (persistence group), and 43% (desistance group: 28 boys and 5 girls) were no longer gender dysphoric.
Most children with gender dysphoria will not remain gender dysphoric after puberty. Children with persistent GID are characterized by more extreme gender dysphoria in childhood than children with desisting gender dysphoria. With regard to sexual orientation, the most likely outcome of childhood GID is homosexuality or bisexuality.Psychosexual outcome of gender-dysphoric children.
This is why the Endocrine Society currently recommends puberty blockers until age 16 instead of cross-sex hormones.
The study the WSJ is likely referring to is this one from 1985:
Two groups of males were evaluated on parameters of gender identity, initially in boyhood and later in adolescence or young adulthood. One group was composed of 66 clinically referred boys whose behaviors were consistent with the diagnosis of gender identity disorder of childhood. The other group consisted of 56 volunteers selected on the basis of demographic matching. Two-thirds of each group were reevaluated for sexual orientation; 30 of the 44 who previously had shown extensive cross-gender behavior and none of the 34 in the comparison group were bisexually or homosexually oriented.Gender identity in childhood and later sexual orientation: follow-up of 78 males
I don't have access to the full study, but the abstract doesn't seem to directly support the claim of the WSJ, as the study is about the percentage of boys who match the GID criteria and grow up to be bisexual or homosexual, not how many of them still match the GID criteria.