Don't remove excess glue with a wet rag! That's what the cabinetmaker did to this face frame.
The principle is simple: the water dilutes the glue and spreads it around, so you've just created an area of finish-resistant wood, which won't show until you apply finish.
This was on a set of cabinets I finished for a cabinetmaker who wasn't willing to apply anything other than quick-dry poly. The homeowner wanted a stain somewhere between rosewood and cherry, and so I was called in to apply a water-based dye stain (a mixture of cherry, rosewood, and blue) under a clear coat of acrylic. The cabinets were soft maple, and looked great until I put the stain on. I took this picture as a way to help explain to the cabinetmaker why the job would take a week longer than he'd planned - - I had to sand everything he had built until the dye would take uniformly, which was a trial-and-error process. The finish-resistant area is invisible, though usually it's palpable, with the affected area feeling slightly slipperier than the bare wood. He must have used a sopping rag to wipe up his squeezeout, because the diluted glue had soaked deep into the wood.
I look back upon that extra week with those 30-plus cabinets and doors as one of my most miserable experiences in woodworking.
The same principle is at play when you pretreat wood before staining it. Some use a thin coat of shellac for this, others buy various proprietary concoctions, but I have had great success with glue size. Mix 9 parts water to one part ordinary PVA glue (yellow or white wood glue), and wipe it on as the first coat of finish. Keep the whole surface saturated for a minute or so, then wipe it all dry. This tends to raise the grain a bit, so next scuff-sand with 220 grit, then apply your water-based stain. Add a coat of Zinsser's Seal Coat, then 2 coats of Ceramithane, and you have something to rival a cabinet factory finish, especially on figured maple: