Thousands of residents have been evacuated and a state of emergency has been declared in Kursk after Ukraine launched a rare incursion into the Russian border region earlier this week.

Russia's defense ministry said Thursday that its forces in Kursk continued to "destroy armed formations" from Ukraine. Posting on Telegram, the ministry claimed that Ukraine's losses since the start of the Kursk operation amounted to 660 troops and 82 armored vehicles including "eight tanks, 12 armored personnel carriers, six infantry fighting vehicles, 55 armored fighting vehicles, and one counter obstacle vehicle."

The number of military vehicles mentioned is far higher than the ministry's initial figure that 11 tanks and around 20 armored vehicles were involved in the initial incursion that began Tuesday.

CNBC was unable to verify the information in the latest post. Neither Russia nor Ukraine publish fatality figures and both sides have a vested interest in exaggerating each other's losses. After initially downplaying the bold Ukrainian incursion into Kursk, Russia is also seen to be wary of dismissing the apparent new offensive on its territory.

On Wednesday, acting regional Governor Alexei Smirnov said the "operational situation" in the border area remained "complicated."

"In order to eliminate the consequences of enemy forces entering the region, I have decided to declare a state of emergency," Smirnov said in a Google-translated update on Telegram, adding that drone and missile attacks had continued overnight.

On Thursday, acting Deputy Governor Andrei Belostotsky said around 3,000 people were evacuated from shelled areas of the region, with four people killed, according comments reported by Russian state news agency Tass.

He also said Ukrainian units "had not advanced a single meter," in comments published by Russian state news agency Ria Novosti. CNBC was not able to verify the claims.

Russia appeared to be caught off-guard when the incursion started on Tuesday with Russia's Defense Ministry appearing to play down its significance, attributing it to a "sabotage and reconnaissance group." At that point, it said 300 Ukrainian troops had crossed over the border with tanks and armored vehicles, entering the country near the town of Sudzha, around 400 miles southwest of Moscow. Kursk lies across the border from Ukraine's northeastern Sumy region.

Russian President Vladimir Putin on Wednesday accused Ukraine of carrying out "yet another large-scale provocation" at the Russian border, amid ongoing fighting between Ukrainian units and Russian forces sent to defend the area.

Russia's Defense Ministry claimed Wednesday that its forces were "destroying armed formations" using air and missile strikes, along with artillery.

Valery Gerasimov, the chief of Russia's general staff, later told Putin that Russian forces had halted an offensive undertaken by 1,000 Ukrainian soldiers — a troop estimate far higher than the original figure supplied by the country's Defense Ministry.

"The enemy's advance deep into territory in the Kursk direction was stopped by the actions of the units covering the state border together with border guards and reinforcement units, with air strikes, missile and artillery fire," Gerasimov said in televised comments reported by Reuters.

The Ukrainian authorities have not made any public comment on the incursion. CNBC has requested further information from the Ukrainian Defense Ministry.

Ukrainian presidential advisor Mykhailo Podolyak on Thursday suggested that Russia's aggression had prompted the latest operation.

"The root cause of any escalation, shelling, military actions, forced evacuations, and destruction of normal life forms, including within (Russia's) own territories like Kursk and Belgorod regions, is solely Russia's unequivocal aggression," Podolyak posted on social media platform X.

U.S. State Department spokesperson Matthew Miller told reporters on Wednesday that the U.S. had not been forewarned of a Ukrainian incursion, but did not rule out that an offensive took place.

"We did not [know in advance about the Kursk operation]. But it's not unusual for the Ukrainians not to notify us of their exact tactics before they execute them. It's a war that they're conducting. We provide them with equipment. We provide them with advice. But when it comes to the kind of day-by-day tactics that they carry out, the day-by-day strikes that they take ... it's appropriate for them to make those decisions," he said in a press briefing.

Miller said that the U.S. was currently "in communication with the Ukrainians on this particular operation" but that it was inappropriate to comment on "what kind of operations they're conducting and what their goals are. That's appropriate that they speak to that publicly, not us."

The U.S. State Department spokesperson further said Putin's description of the incursion as a "provocation" was "a little bit rich ... given Russia violated Ukraine's territorial integrity and sovereignty" when it has illegally occupied Ukrainian territory since 2014, beginning with the annexation of Crimea, ahead of the large-scale invasion of February 2022.

Ukraine's strategy and reasons for carrying out a border raid remain unclear, although some defense analysts have commented that the operation may be an attempt to force Russia to redeploy forces away from eastern Ukraine, where a Russian summer offensive has been gaining ground.

Analysts at the Institute for the Study of War estimated Wednesday that Ukrainian forces had made confirmed advances of up to 10 kilometers — or around six miles — into the Kursk region "amid continued mechanized offensive operations on Russian territory."

"The current confirmed extent and location of Ukrainian advances in Kursk Oblast [region] indicate that Ukrainian forces have penetrated at least two Russian defensive lines and a stronghold," the ISW said in an analysis.

The institute said "geolocated footage published on August 6 and 7 shows that Ukrainian armored vehicles have advanced to positions along the 38K-030 route about 10 kilometers from the international border."

The ISW said that the Kremlin's response to Ukrainian offensive activities in Kursk "has so far been contradictory, as Russian officials are attempting to balance presenting the effort as a notable Ukrainian escalation with avoiding overstating its potential implications and risking domestic discontent."

"The Kremlin risks, however, discrediting itself among certain communities by seemingly dismissing the significance of the attack by framing it only as a 'provocation."'

The ISW noted that a number of Russian military bloggers, often critical of the war and Russia's tactics and strategy, have heavily blasted Russia's military command for not detecting preparations for Ukrainian offensive operations into Kursk or preventing the initiative.

The deputy chairman of Russia's Security Council and well-known hawk Dmitry Medvedev called on Russian forces to crush the Ukrainian units in Kursk.

"We need to learn a serious lesson from what happened and do what Chief of the General Staff Valery Gerasimov pledged to the supreme commander-in-chief to do, that is, resolutely defeat and crush the enemy," he stated on Telegram, according to a Google translation.