It depends on what type of credit the specific GSE is facilitating. GSEs are “Government Sponsored Enterprises” that are set up to facilitate the flow of credit to the private sector by guaranteeing loans and/or purchasing loans on the secondary market. Fannie Mae, Freddie Mac and Farmer Mac generally focus on mortgages, while Sallie Mae focuses on student loans. Yes, all bonds sold by GSEs are likely callable at any time if pre-payments on the underlying loans reach a particular threshold.
Prior to the financial crisis, the average duration for a 30-year mortgage was only about 10 years, generally because of refinancing or sale of the underlying property. With the extremely low-rate mortgage loans that were issued since the financial crisis, there is less incentive to refinance, so the average age of loans has been increasing, making calls of mortgage-backed bonds less likely.
AJ