So you want constant $P,T$ while $V$ presumably changes due to the moving piston. But if $V$ is increasing (and if external $P\neq0$, i.e., not a so-called "throttling process"), then the gas is doing work on its surroundings. So if the process is adiabatic, then that work energy has to come from within the gas itself, whereby $P,T$ has to change (a la first law) correspondingly. Otherwise, if you want the gas's $P,T$ constant while it's expanding, you'd have to supply outside energy to make up for the energy lost to the piston head, and the process won't be adiabatic.

So you want constant $P,T$ while $V$ presumably changes due to the moving piston. But if $V$ is increasing (and if external $P\neq0$, i.e., not a so-called "throttling process"), then the gas is doing work on its surroundings. So if the process is adiabatic, then that work energy has to come from within the gas itself, whereby $P,T$ has to change (a la first law) correspondingly. Otherwise, if you want the gas's $P,T$ constant while it's expanding, you'd have to supply outside energy to make up for the energy lost to the piston head, and the process won't be adiabatic.

edit Re your "what happens in particles?" part of the question, just consider a gas molecule colliding with the piston. If the piston's not moving, then, ideally, the collision's elastic, and macroscopic $P,V,T$ aren't affected. But in the non-equilibrium situation where the piston's expanding, some momentum's transferred from the molecule to the piston, resulting in corresponding changes to $P,V,T$.

So you want constant $P,T$ while $V$ presumably changes due to the moving piston. But if $V$ is increasing (and if external $P\neq0$, i.e., not a so-called "throttling process"), then the gas is doing work on its surroundings. So if the process is adiabatic, then that work energy has to come from within the gas itself, whereby $P,T$ has to change correspondingly. Otherwise, if you want the gas's $P,T$ constant while it's expanding, you'd have to supply outside energy to make up for the energy lost to the piston head, and the process won't be adiabatic.

So you want constant $P,T$ while $V$ presumably changes due to the moving piston. But if $V$ is increasing (and if external $P\neq0$, i.e., not a so-called "throttling process"), then the gas is doing work on its surroundings. So if the process is adiabatic, then that work energy has to come from within the gas itself, whereby $P,T$ has to change (a la first law) correspondingly. Otherwise, if you want the gas's $P,T$ constant while it's expanding, you'd have to supply outside energy to make up for the energy lost to the piston head, and the process won't be adiabatic.