Changing a rule input variable in a child interface changes it in the parent but SAIL in local variable doesn't?

Sorry, but I do not understand what you are doing here. Please help me.

Why do you store a UI component into a local?

Instead of updatedictionary(), I suggest to use a!update().

"does" or "doesn't"?  It sounds like you meant to say "doesn't" but it's a little ambiguous.

If you declare interface components in a local variable, strange things can and will happen sometimes.  You CAN (usually) get it to work, however, but I believe the code in the local variable must be manually declared using a Refresh Variable set to "REFRESH ALWAYS" behavior.

local!sailInterfaceDropdown: a!refreshVariable(
  value: a!sideBySideLayout(
    item: rule!nestedDropdownfield(
      choiceLabels: local!listOfYesNo
      choiceValues: local!listOfYesNo
      value: index(
        local!madeFromCDT,
        "arbitraryField"
        saveInto: a!save(
          local!madeFromCDT,
          updatedictionary(
            dictionary: local!madeFromCDT,
            fieldsAndValues: {arbitraryField: save!value}
          )
        )
      )
    )
  ),
  refreshAlways: true()
)

Additionally, since this manually references the earlier local variable, it's not clear what would happen if, for instance, you then try to reference this local variable multiple times within an interface.  It might just break, or do something even more unexpected.  Generally I avoid doing this except for when I have static UI elements I want to use several times on a form without having to manually declare them everywhere, but even then they only reference generic external variables (like simply saving a single value into a single rule input, like a submit button or similar).