"Decay" is just the name given to an interaction where one particle goes in and two or more particles go out.
The rule in quantum mechanics is "anything not forbidden is compulsory" – that is, any process (decay or otherwise) can happen unless it violates a conservation law. As a result, most particles, whether fundamental or composite, do decay.
The exceptions are particles for which there's literally no set of outputs you can choose that doesn't violate some conservation law. For example, electrons can't decay because the decay would have to conserve electric charge – so at least one output would have to be charged – and would also have to conserve mass/energy – so the total mass of all the outputs would have to be no larger than the electron's mass – and this is impossible because there are no electrically charged particles with lower mass. So it's stable not because it's elementary but because everything that could make it unstable is forbidden.
The muon, which is also fundamental and is almost identical to the electron except for its mass, can and does decay, because the higher input mass means that you can find outputs that conserve mass while also satisfying all other constraints.
The proton, which is not a fundamental particle, can't decay because it's the lightest particle with another conserved property called baryon number.
But, again, the particles that can't decay are the exceptions. As a rule almost everything can and does, and this applies to fundamental and composite particles alike.