What the evidence says
The claims here come from two multi-site descriptive studies and one Boston lottery study. They address whether racial-differential charter patterns seen in North Carolina appear elsewhere. The evidence is uneven: it is strongest on who transfers and thinnest on achievement effects by race.
A seven-site RAND study (Zimmer, 2009) tracked students as they moved into charter schools. African American students who transferred moved to schools with higher concentrations of African American students in five of seven locales. The study reports that the effect on racial integration differs by place. Transfers to charter schools tend to marginally reduce racial integration in Philadelphia and in Texas, and marginally increase it in Chicago. Frankenberg (2011), summarizing the same study, reports that Black students moved to more segregated schools in five of seven locations. In Chicago and Milwaukee they attended slightly less segregated charter schools than they would have in public schools. Both of those traditional systems had very low percentages of White students. These findings are associational. The claims do not say whether the pattern reflects student preference, charter location, or capacity.
The same RAND study compares prior achievement of transfer students with same-race peers who stayed in traditional public schools. White students entering charter schools had higher prior achievement than their white peers in four of seven sites, and in one other site in one of two subjects. Lower prior scores for charter entrants appear in five of seven sites among African Americans and in four of seven sites among Hispanics. The direction of selection therefore differs by race in most sites. These are descriptive comparisons of prior scores, not effects of charter attendance.
On achievement effects by race, the evidence is thin. In Los Angeles Unified and San Diego, Frankenberg (2011) reports that charter student achievement did not vary substantially across the race or language status of charter students. In Boston, a lottery-based study finds that middle-school students who win the charter lottery score almost 0.2σ higher on ELA and 0.4σ higher in math. The authors report that separate estimation by race, not reported in detail, suggests the effects apply equally to Black and White students. Boston's charter schools also serve a high proportion of Black students, even relative to the majority non-white district. The Boston race-specific result is not shown in the source, so its magnitude is unsupported here.
No claim here covers a replication of North Carolina's specific racial-differential charter effects. The sources describe other places, not a direct test of the North Carolina result.
Where the evidence splits
The integration findings differ by location. Charter transfers are associated with marginally less racial integration in Philadelphia and Texas, and marginally more in Chicago. Frankenberg (2011) adds that Chicago and Milwaukee are the two sites where Black students ended up slightly less segregated. The sources attribute the Chicago and Milwaukee result partly to the very low percentage of White students in both traditional systems. They do not test that explanation, and they do not explain the other differences between sites.
The Boston lottery study and the RAND studies do not measure the same quantity. The RAND claims describe who transfers and the peer environment. The Boston claims estimate achievement effects of winning a lottery. The claims do not conflict on a shared outcome.