Wouldn't this put weird stresses on the film though and encourage it to curl etc whilst drying?
No.
There is just enough tension to keep it from sagging.
And the modern film we use is designed to be able to withstand a lot more than this - particularly in commercial processors and high speed motor drives.
The old Kodachrome machines ran three miles of spliced together films through the machines at continuous high speed.
And current motion picture film processors do the same with Vision films and 5222 black and white film, which is extremely similar to still films on acetate base. The Estar based films are even more robust.