Ukraine’s lightning offensive into several Russian border regions is designed to persuade Moscow to engage in “fair” talks about its war in Ukraine, an aide to Volodomyr Zelenskiy has said.

The Ukrainian presidential adviser Mykhailo Podolyak wrote on the Telegram messaging app: “We need to inflict significant tactical defeats on Russia. In the Kursk region, we clearly see how the military tool is objectively used to convince the Russian Federation to enter into a fair negotiation process.”

Both sides were pushing ahead with their offensives on Friday, as Russian forces edged closer to the key city of Pokrovsk in Donetsk and Ukraine appeared to be consolidating its gains in Russia’s Kursk region.

Russia has accused Nato and the west more widely of aiding the Ukrainian incursion, including by permitting the use of western-supplied kit. But British officials said Ukraine was entitled under international law to use British-donated equipment in operations, including within Russia.

“There has been no change in UK government policy, under article 51 of the UN charter Ukraine has a clear right of self-defence against Russia’s illegal attacks, [and] that does not preclude operations inside Russia,” the Ministry of Defence said. “We make clear during the gifting process that equipment is to be used in line with international law.”

It appears, however, there has been no change in the UK’s refusal to allow Ukraine to use British-supplied Storm Shadow missiles against targets inside Russia, suggesting a delicate balancing act.

The US so far has also deemed the incursion a protective move in which it is appropriate for Kyiv to use US equipment, officials in Washington said. But they also expressed worries about complications as Ukrainian troops push further into enemy territory.

One US official, speaking on condition of anonymity, told Reuters that if Ukraine started taking Russian villages and other non-military targets using US weapons and vehicles, it could be seen as stretching the limits Washington has imposed, precisely to avoid any perception of a direct Nato-Russia conflict.

Ukraine has said that one of the aims of its current incursion into Russia is to counter ongoing artillery and missile fire into Ukraine and create a buffer zone.

Footage posted on Russian social media has purported to show western-supplied equipment – including a British-supplied Challenger 2 main battle tank – destroyed or captured during the Kursk offensive, although the tank depicted in the footage appears to be a Soviet-era T-64. Unverified reports have said that Challengers may have been used during the operation.

Russia’s defence ministry has also published footage that it said showed a Russian drone destroying a US-made Stryker armoured combat vehicle in Kursk.

On Friday Russian forces said they had destroyed a Ukrainian reconnaissance and sabotage unit in Kursk that was armed with weapons from Nato countries, the state-run media agency RIA reported, citing unidentified security sources.

“Samples of small arms manufactured by the United States and Sweden have been seized at the liquidation site of a Ukrainian sabotage group near the village of Kremyanoe in the Kursk region,” RIA cited a Russian security official as saying.

Ukraine’s attack into Russia began on 6 August when thousands of Ukrainian troops crossed Russia’s western border, in a major embarrassment for the Russian military.

The US and other western powers, eager to avoid direct confrontation with Russia, said Ukraine had not given advance notice and that Washington was not involved – assertions dismissed by the hawkish Kremlin aide Nikolai Patrushev in an interview with the Izvestia newspaper.

“The operation in the Kursk region was also planned with the participation of Nato and western special services,” he was quoted as saying, without offering any evidence. “Without their participation and direct support, Kyiv would not have ventured into Russian territory.”

In Moscow, one lawmaker said the Ukrainian incursion and the presence of western military equipment on Russian soil had brought a third world war a step closer, echoing a line much repeated by the Kremlin over the past two years.

So far the sluggish Russian response has seen mostly irregular units redeployed.

Footage geolocated by the Institute for the Study of War placed Ukrainian troops just under 30km (18.6 miles) from the international border in Kursk. Although the Russian ministry of defence claimed it had cleared some settlements of Ukrainian forces, Russian military bloggers suggested that fighting was still continuing.

On Thursday a number of Ukrainian news outlets identified several units of more than 100 recently captured Russian prisoners of war, suggesting a mix of regular and irregular forces including Chechen fighters.

Fighting also appeared to be continuing at a border crossing into the Belgorod region – a second line of Kyiv’s advance – with conflicting reports over the status of the fighting.

While the Ukrainian attack has revealed weaknesses in Russia’s defences and changed the public narrative of the conflict, Russian officials said what they cast as a Ukrainian “terrorist invasion” would not change the course of the war.

Russia has been advancing for most of the year in the key eastern sector of the 1,000km (620 mile) frontline and has vast numerical superiority.

Reuters and Associated Press contributed to this report