Sophisticated companies understand they lose good candidates with this type of application process. This can mean the company is in trouble or it's a sweatshop environment. If you are prey to constant offers and counter-offers, eventually no one will trust you to show up for your new start date. Make a checklist for yourself and walk through it. If you're any good, your company should make you a counter offer.
Nobody wants to lose a good employee.
If you stay just because of a raise, you're likely to remain recruiter bait and leave your current company within the year. If you walk away, you'll feel better and so will everyone else. Don't hide information from the recruiter about other interviews and packages but work honestly with that middleman. If you have requirements, let the recruiter help you negotiate. If you burn your bridges with a recruiter, this will surely boomerang somewhere down the line.