Lead - Pronounced - Led.
Barrels are made from steel which is a much harder alloy than Soft Led.
As above that I posted.....Joe Friedrich won an ARA sanctioned benchrest match using a gun with a barrel that had more than 100,000 .22 rimfire rounds passed through it. As far as I know he is still using that gun years later, among others he has in his collection.
In summary a barrel shooting a soft Lead projectile , if taken care of , can last a lifetime for most shooters. Now once you step up to powder burners/firearms and larger much faster calibers that produce a lot of heat, this all changes quickly and you run into things like throat erosion, and a bevy of other issues after extended use. Airguns are different. Conversely, you are shooting rapidly decompressed air which is cold. Instead of getting hot, the barrel cools.
Once more, In summary, no, shooting slugs will not harm a barrel of an airgun. ( unless they are dirty and covered in grit and whatnot else. If you drop one on the ground, throw it away. Don't put it through the barrel. Especially in a .22 rimfire firearm/powder burner. Rimfire cartridges have a form of lubrication on the led bullet. Once it hits the ground , whatever dust or grit that is on the ground is picked up and sticks to the lubricant. Don't shoot it. Just throw it away.