The Ukrainian defense ministry has reported further ceasefire violations in the east, after a day of heavy weapons fire Saturday.
The ministry said that in the first 11 hours of Sunday, "20 incidents of ceasefire violation by the Russian occupation forces were observed, including 18 incidents when the Russian occupation forces utilized weapons prohibited by the Minsk Agreements."
The Minsk II agreement led to a shaky ceasefire between the Ukrainian military and Russian-backed separatist forces, and bans heavy weapons near the line of contact between the two sides.
Ukraine said it recorded a total of 136 ceasefire violations on Saturday.
The Ukrainian Border Guards said that because of the shelling one crossing point for international humanitarian organizations, Shchastia, at the Line of Contact had been closed since 8:00 a.m. local time Sunday. A UNHCR convoy that used the crossing point Friday said it had been caught in crossfire.
Some residents of Donetsk -- which is controlled by pro-Russian separatists -- reported heavy shelling early Sunday. One woman contacted by CNN said she and her children wanted to move closer to the city center because of shelling in her district, Abakumova.
I can't survive another night like this. I'm really scared," she said.
It's unclear where the shelling originated. The authorities in the breakaway republics persistently claim shelling by Ukrainian forces, who in turn regularly deny firing artillery across the front lines.
The woman also told CNN that on Saturday her neighbors had left for Russia and had been accommodated somewhere near Rostov-on-Don in a tent encampment.
The Russian authorities say that more than 40,000 people have arrived in Russia after being evacuated from the self-proclaimed Donetsk and Luhansk People's Republics, according to the acting head of the Ministry of Emergency Situations Alexander Chupriyan.
"They are located mainly in the Rostov region," Chupriyan said.
CNN is unable to verify independently the numbers crossing into Russia.
The governor of the Rostov region, Vasily Gobulev, Saturday announced a state of emergency in the region to accommodate the influx.
The evacuation of civilians was ordered by the leaders of the breakaway republics on Friday, when they alleged that Ukrainian armed forces were planning an offensive against the regions. Ukraine has consistently denied any plans to attack the regions, which comprise some 7% of Ukrainian territory.
Russia's Investigative Committee says it is beginning an investigation into media reports of fatalities due to shelling in Luhansk region. In a statement, the committee said: "The media reported that this morning the Armed Forces of Ukraine made an attempt to attack the positions of the People's Militia of the self-proclaimed Luhansk People's Republic, while, according to preliminary data, there are casualties among the civilian population."