One set of documents discusses the likely Russian reaction to the ship’s passage through Ukrainian waters off the Crimea coast on Wednesday.
Another details plans for a possible UK military presence in Afghanistan after the US-led Nato operation there ends.
The government said an investigation had been launched.
The Ministry of Defence (MoD) said it is investigating “an incident in which sensitive defence papers were recovered by a member of the public”. The MoD employee concerned reported the loss at the time, it said.
The documents, almost 50 pages in all, were found in a soggy heap behind a bus stop in Kent early on Tuesday morning.
A member of the public, who wishes to remain anonymous, contacted the BBC when he realised the sensitive nature of the contents.
The BBC believes the documents, which include emails and PowerPoint presentations, originated in the office of a senior official at the MoD.
The documents relating to the Royal Navy’s Type 45 destroyer, HMS Defender, show that a mission described by the MoD as an “innocent passage through Ukrainian territorial waters”, with guns covered and the ship’s helicopter stowed in its hangar, was conducted in the expectation that Russia might respond aggressively.
On Wednesday more than 20 Russian aircraft and two coastguard ships shadowed the warship as it sailed about 12 miles (19km) off Crimea’s coast.
Moscow’s defence ministry said a patrol ship fired warning shots and a jet dropped bombs in the destroyer’s path but the UK government rejected this account, denying any warning shots had been fired.
The mission, dubbed “Op Ditroite”, was the subject of high-level discussions as late as Monday, the documents show, with officials speculating about Russia’s reaction if HMS Defender sailed close to Crimea.
“What do we understand about the possible ‘welcome party’…?” asked an official at Permanent Joint Headquarters (PJHQ), the UK’s tri-service headquarters at Northwood.
Recent interactions in the eastern Mediterranean between Russian forces and a Carrier Strike Group led by HMS Queen Elizabeth had been unremarkable and “in line with expectations”, the document said.
But officials knew this was about to change.
“Following the transition from defence engagement activity to operational activity, it is highly likely that RFN (Russian navy) and VKS (Russian air force) interactions will become more frequent and assertive,” one presentation warned.