Whether a musician has lost or sold the rights to his music (the Beatles were in that situation for most of their career, although Paul McCartney has regained some publishing rights to some of his songs), someone or some entity has those rights and absent fair use (see below) no one is allowed to play the copyrighted music in a public setting without either permission or paying some kind of royalty.
Even when bands play covers or DJs play music, the venue itself is either paying royalties or has some kind of blanket arrangement whereby something is paid to the copyright holders.
When it comes to politicians playing music at their events, they too need to pay for use of the music and if the copyright holder such as the band doesn't object, nothing extraordinary happens, some kind of compensation is passed along if the holder requires it, but if the copyright holder objects, the politician is supposed to not play or stop playing the musician's music. For the most part this does not become an issue, but with someone like Grump who gets people fired up one way or another, there are some musicians who have asked him to stop playing their music and have gone as far as sending cease and desist letters and threatening legal action.
And some, have taken it all the way to lawsuits.
More and more artists want Trump to stop using their music. They face a costly fight
By the way, a Grump defense (that will probably fail) and over arching principle that defeats copyright infringement is called
fair use, where someone who uses copyrighted material in a non-commercial manner, for example for educational use, and not for profit, doesn't need a license (doesn't need permission) doesn't need to pay for its use. But Grump is using the music to promote his campaign which is far from fair use.
Still, if you pay attorneys enough (and Grump has lined his pockets with quite a bit of his supporters funds, such that he just burns OTHER people's money not his for such things), they will come up with creative arguments that will make it very costly to beat them, even if the end result is inevitable.