In your examples, the -ing forms are seen as nouns requiring possessive determiners, such as my or his. They can, however, also be seen as non-finite verbs, requiring the accusative forms of personal pronouns, such as me or him.
The choice depends on the emphasis which the speaker or writer wants to give. The ‘Longman Student Grammar of Spoken and Written English’ explains that:
. . . when the possessive alternative is used, it focuses attention on
the action described in the –ing clause. In contrast [the
non-possessive] form puts more emphasis on the person doing the
‘The Cambridge Guide to English Usage’ makes the additional point that:
The choice of the possessive my [for example] makes the sentence
rather formal, while the use of the object pronoun me is acceptable in
most everyday kinds of writing.