the design of the reduced thickness top section is unconventional for U.K. tumblers, although I have a feeling that rims like these did have some vogue in the late C19 and perhaps early C20, in Europe. I'm thinking these are possibly from your side of the pond - perhaps pieces shaped thus were for some specific purpose, or maybe just standard States tumblers for whiskey/brandy. Is it this particular design that makes you suggest 'old fashion'?
You might try the internet Replacements sites, but assuming no one recognizes them you could be in for a long wait - sorry to be pessimistic. However, fingers crossed that someone may recognize the design.

Hope you won't mind me offering suggestions for future wording of posts, but it's usual practice to word your request differently from your subject heading, and to show the nature of your request a little more clearly.
