Special collections not recorded on institution's ledgers

We received some questions from a chartered museum about weeding from the museum's internal research library:

• What legal considerations govern weeding and/or deaccessioning the materials from a museum's library?
• What if the museum’s library isn't sure how it acquired the materials?
• What might the legal ramifications be if the institution were to transfer select items from their library to a historical society more in line with their content and origins?
• Does the museum have the legal right to sell the materials at auction?