The exclusive world of luxury retailing is, in some ways, the last bastion of an old-school approach to shopping.

Hermès does not sell its signature Birkin handbag online.  Chanel, beloved for its tasteful purses and ladylike clothes, only makes its fragrance and beauty products available for online purchase. And you won’t find any $3,600 Celine handbags for sale on the brand’s Web site, or for that matter, on the sites of department stores such as Bergdorf Goodman.

Even some of the luxe labels that do sell their goods online have sometimes been cautious about trumpeting their brand aggressively on Instagram or Facebook, determined to simply let their chic Madison Avenue window displays (or maybe their ads in the pages of Vogue and Elle) do the talking.

This caution around making the leap to digital, experts say, has been deliberate:  In part, it’s about preserving exclusivity by ensuring that your high-end goods are hard to come by. These retailers may also want to draw a bright line between their authentic goods and the knock-offs of their products that have become prolific on online marketplaces.

But, a new report from McKinsey & Co. finds that it is a mistake for luxury brands to do anything other than wholeheartedly embrace the online channel as a crucial opportunity for ringing up sales and influencing in-store purchases.

“It’s a quite traditional industry where customer experience has always been paramount,” said Nathalie Remy, one of the report’s authors. “But today, the definition of customer service for the 21st-century shopper is not exactly what it used to be.”

McKinsey found that nearly all of the growth in the global luxury goods market in 2014 came from e-commerce purchases.  While digital purchases made up just 2 percent of luxury sales in 2009, that figure has ticked up to 6 percent, McKinsey found. And in studying the shopping “decision journeys” of 7,000 luxury shoppers across the globe, the consulting firm founded that three out of four luxury purchases are now influenced by digital interactions, including Web browsing or trawling social media.

McKinsey also pokes holes in the idea that perhaps luxury brands can be less focused on their Web presence because they often target older buyers. The report found that among the luxury buyers, Baby Boomers spend just as much leisure time online as millennials do and own nearly as many mobile devices.   In other words, affluent Boomers are just as ripe targets for e-commerce as their younger counterparts.

The smartphone, Remy said, “is really the trigger that has radically shifted the behavior.”

The report found that for every image a luxury brand posts on its official Instagram account, there are an average of 10,000 more posts from consumers about the brand. Many luxury retailers have kept only a quiet presence in the online channel to keep tight control over their brand images.  But this finding perhaps illustrates that this strategy is not helping them retain control, but is actually helping them lose influence over the brand by ceding their storytelling to others.

If luxury retailers do indeed move more forcefully into the online channel, Remy said they will have some unique challenges that are not necessarily shared by their mass market counterparts.

Because luxury brands produce high-value goods in very low quantities, it “makes the e-commerce supply chain a nightmare,” Remy said.  In other words, it’s impractical to have scores of distribution centers dotting the globe when you don’t have a huge volume of goods.  And so these luxury brands will have to think strategically about the logistics of getting their wares into the hands of these digital-minded shoppers.