The usual commercial method of putting markings on textiles is with silk-screening, so named because originally a silk mesh matrix was used to hold the "masking" material, and a paint was wiped/rubbed through the unmasked areas. Vendors who sell shirts at events use a hot iron to "melt the paint" into the fabric, but dab-and-dry also works reasonably well with appropriate paints. (One vedor at a venue we visit has recently been using the "iron on trasfer paper" that you print on your inkjet printer and then squash into the paper with a hot iron. Results are less satisfying than screening.)

As little kids (6 - 9 year old Cub Scouts in a previous century) we made shirts and flags and the like with the usual "propaganda slogans of the day" on them using what amounted to "waxed paper stencils" hand cut by a rare "coordinated volunteer" with the unusual ability to "draw something with a knife." The paint needed to dry, but didn't take long (a few seconds) to get to the point where you could handle the article with a little care if the paint wasn't applied too thickly, although "full dry" was a matter of at least minutes.

For making the same marks on many pieces, a rubber stamp gives a similar effect. With stamping, applying minimal amounts of the "ink" in thin layers with a little pressure can give a very short "drying time" and minimal "bleed" even with ordinary office store inks. There are (or were once-upon-a-fabled time) stamping inks specifically for textiles, but I haven't seen ads for those inks in half a century. (There is more "technique" to using a rubber stamp successfully for unusual purposes than one might imagine, so a little practice would be advisable before trying to do a lot of it in public.)

Of course stencils (silk screen) and stamps limit you to the designs you have on a stencil/stamp.

Michaels will have "paint pens" you can use to "draw" on just about anything. The paint in most is similar to the latex/acrylic paint you'd use a brush with and does need at least a few seconds to dry. It's also difficult to draw a line less than about an eighth inch wide with the kinds of pens most frequently seen. They're about like using "typewriter correction fluid" (WhiteOut?) with a felt tip pen, but with a choice of colors. They should also have the "fabric markers" (most often just black) but it's hard to guess how much bleed you'd get on fine silk, so "test is best" would be the rule.

The ordinary office store "Sharpie" might work for what you want to do. They're not indicated as "for textiles" but they do come in "permanent" and "less-permanent" types so you want to watch which you get.

John