You want to be become a non-employee, remote collaborator, abroad.
Does your company allow you to these aspects independently? Do they have people working as contractor? Do they have people working remotely? Do they have people working abroad? All of this in a role similar as you?
There are a number of reasons (*) an employer does not want to deal with all consequences of an employee working abroad. As non-employee abroad you will need to deal with yourself; for the countries that you are working from.
If you want them to go along with this, you will need to convince them that you are/will be covering all of this yourself.
Think: contractual agreements as non-employee, right to work, all practical aspects of working 'very' remote (connectivity, availability, accessibility, confidentiality, ... ), taxes ( theirs and yours), contributions ( theirs and yours) for healthcare + social security.
Check your contract and employee handbook for aspects that your company find important.