Delta, for one, is unapologetic.

“Delta reserves the right to determine who it does business with and where and how its content is displayed,” a company spokesman said in an email, adding that the carrier continued to work with a limited number of online partners.

Both sides in this tug of war have a case to make.

“The airlines absolutely want fliers as much as possible to come to their own website,” Mr. Quinby said

There are two financial advantages: Airlines want to be able to offer their own fare packages, bundles and add-ons like extra legroom and in-flight entertainment.

“The more that the airlines can get the fliers to book on their own sites, the more opportunity the airlines have to engage with those travelers,” Mr. Quinby said.

In the industry, that is called the upsell. “Where a significant portion of Delta’s profits come from is ancillaries after you buy the basic transportation,” said Max Rayner, a partner at Hudson Crossing, a travel industry consulting company. “That’s one big reason for Delta to be hellbent on denying access.” Last quarter, for example, Delta’s extra revenue from everything from baggage fees to merchandising rose 27 percent — an extra $50 million.

Airlines also want to avoid paying fees of around $5 to $12 a ticket when fliers make third-party bookings. Lufthansa said that its new fee was being levied only on booking channels where the carrier’s own costs have gone up.

The third-party players, for their part, want to present a transparent website that lets travelers make an apples-to-apples comparison on flights. At some websites, airlines are a loss leader. The sites make money by earning commissions from hotel bookings, but the flights help lure the customer in the first place.