Hey, Mitzio,
Both, and pretty much any similarly sized stroller, will unfortunately suffer on cobblestones and bigger bumps, so be ready for that. The small frames and small wheels just can't perform better on those. For car trips and less demanding walks, I would probably go for the roomier sat here, which, to me, the Beezy has (the Influencer is nice but feels shorter in the backrest area). However, the larger wheels of the Influencer XL will perform a tiny tad better on cobblestones, so if that were a priority, the Influencer XL would be the way to go. It is, however, a bit larger after folding, which would mean you might not be able to take it on board the plane as cabin luggage.
I personally would, however, probably go for the Joolz Aer+ which, to me, is one of the sturdier models in the category, and also one of the largest seat/tallest seat backs from these small-sized buggies. Apart from that, also look at the Cybex Eezy S+2 which is similar to the Beezy but has larger wheels, similar to the Influencer, and a tad larger seat.
In any case, all are reasonable options, so you won't make any big mistake out of what you chose. Not everyday strollers, however, these will do best as second strollers for the car, shopping, easy walks, doctor trips, holidays, etc.