If you mean "in practice it will vary by location because it won't be everywhere provided by the same organization, and thus likely won't be provided everywhere

If you mean "in practice it will vary by location because it won't be everywhere provided by the same organization, and thus likely won't be provided everywhere under the same rules," then I agree with your expectation. If you mean "a basic income, offered by the same organization in multiple places, should be adjusted for cost of living", I disagree. Moving some "extra" funds towards lower cost-of-living areas winds up being a feature - resources are shifted (slightly) away from concentrations of wealth and toward concentrations of people, where those disagree.