Answer first: the receptor is a class B G-protein-coupled receptor signalling mainly through Gs and cyclic AMP, and almost every downstream effect people ask about traces back to where that receptor is expressed rather than to what it does when activated.
Native GLP-1 has a circulating half-life of one to two minutes because dipeptidyl peptidase-4 cleaves the two N-terminal residues. Substituting the position-8 alanine, as the long-acting analogues do, blocks that cleavage and is the single most consequential modification in the class.
Stated carefully, gastric emptying delay attenuates with continued exposure for long-acting agents through receptor desensitisation, which is why the early nausea usually settles while the appetite effect persists.
The position-8 substitution conferring DPP-4 resistance appears in essentially every long-acting agent in the class, which is about as strong a piece of convergent evidence as medicinal chemistry offers.
A receptor being expressed in a tissue does not establish that activating it there matters at therapeutic exposures.
Tissue distribution first, then signalling. Nearly every question in this tag resolves at the first step.
edited 11 Oct 2024 by Dr_Otto_Lindqvist — removed a claim I could not source