the relationship between the heated air particles inside a hot air balloon and the cooler air outside the balloon?

Heating air under the balloon will cause the air inside to expand.

The air inside will have the same mass but more volume.

Heating the air under the balloon will cause the entire balloon to rise.

The heated air temperature from inside the balloon more likely rises since it can't escape its trapped resulting in different density from the cooler air, the cooler air from the outside is a bit denser than the heated.