At present free shipping is a perk that select retailers provide, and there appears to be no 'movement' via either legislature or through trade bodies to make it a universal aspect of online trade. Free shipping will not suit every online business, and any effort to make it universal will probably just result in the price of shipping being added onto the product price.
(Amazon delivery box: Amazon were one of the first online retailers to launch free shipping (supersaver delivery) on all orders, when they reached a price threshold. Eradicating one of the few drawbacks to online shopping: shipping cost)
Historically, free shipping has been provided by businesses which sell products with a low weight ratio to price, such as: jewelry, watches, video games, perfume, aftershave, cd's or dvd's. On the flip side of that coin, its still rare for traders on eBay to provide free shipping for large pieces of furniture, and many furniture sellers are 'collection only'.
Amazon were one of the first (large) online retailers to offer free shipping, but only when an order reached a specific price threshold. The original threshold was £25, which was reduced to £18, and is now priced at £20. Amazon have recently launched their Prime service, which provides free next day delivery on all orders, but becoming a prime member costs £7.99/month.
Most high street retailers have followed Amazon's example, and provide free delivery when an order reaches a price threshold -- there is no universal threshold, Sainsbury's for example, lower their delivery costs when a grocery order reaches over £40. A potential drawback with free delivery (usually standard delivery) is that the delivery time is sometimes stretched to 3-5 days, though free next delivery is sometimes provided on higher priced goods.
In conclusion, it seems highly unlikely that a universal shipping scheme would be implementable, due to the differences between traders, and would impact the sustainability of startup businesses.