Answer:
The original ABA Australian Battodo Association Inc. 
 https://www.australianbattodoassociation.org.au. was founded & incorporated in the State of NSW Australia in 2006 to promote Toyama & Nakamura Ryu Battodo at the Sei-Do-Kan dojo, established in 2001 by the late Nakamura sensei.

As an incorporated association the Australian Battodo Association Inc. is a legal entity and, for others to use the same association's name with different abbreviations, like: IBF; Nakamura Ryu; dot com, must be considered inappropriate and offensive:

IBF- Australian Battodo Association 
Nakamura Ryu - Australian Battodo Association 
australianbattodoassociation dot com   with its hosting-server based in the United States,  

Question:
Who has the authority to grade and, will different associations/federations accept each others ranks?

Answer:
Individuals who hold a licence (shogo title) of renshi-, kyoshi- or hanshi have the authority to grade up to one grade below their own dan-grade

Individuals without such a licence (shogo title) can only grade up to two ranks below their own dan-grade held.

Belonging to an association or federation, individuals   have to comply with the organization's constitution and grading-regulations: clearly defining each grading process.
 
In most cases it's up to each organization to accept or reject a new member's ranking received outside their jurisdiction.