Sometimes they take the owners, somehow cramming the passengers between the chaos of cages that fill the back of their two vehicles.
Patrick, and his owner's 16 other cats, is now settling into a temporary home in Wroclaw, where Lubelska and her three co-volunteers at Ekostraz, the small animal rescue group, are from.
Her eyes puffy with fatigue from the previous day's mission, Lubelska explained that Friday's main target was a German shepherd who recently gave birth to a litter of puppies, but is having trouble breathing and feeding her young.
Getting the dog and her puppies out will be the most daring mission yet.
It will involve, for the first time, travelling about 100km further than Lviv, far closer to Vladimir Putin's invading forces than she has hitherto ventured.
"It's scary, to be honest," she admitted. "But humans are very adaptable. I'm less scared now than I was at the beginning of the week."
If Friday's mission sounds difficult enough, it is nothing compared to what Ekostraz has planned for the weekend.
The team has become aware of a pair of wolves, brother and sister, that they have been told once patrolled the perimeter of a Ukrainian prison and were recently discovered emaciated by an animal shelter, which is itself now running out of supplies.
"We know that the male is called Balto," said Lubelska. "We were told that they used to patrol between two wire fences at a prison. It's a specialist job, so we are going to be collaborating with another organisation to get them home."
On their outbound journeys, the team drops off human medicines in hospitals and clinics, as well as veterinary medicine.
While Lubelska and her co-volunteers have not yet run into any fighting, they have seen war planes streaking low overhead and witnessed the grim preparations for invasion in multiple areas with the making of Molotov cocktails.
She said: "My boyfriend is very worried about me – I don't tell him everything. And my mum thinks I'm just helping out at the border.
"I could be chilling at home designing tattoos. But I'm taking a break from normal life to help Ukraine. It's just that we're specialising in helping Ukraine's animals."
She added: "I suddenly thought 'I can't stand it, I can't just sit here and do nothing'. Pet owners and shelters across Ukraine are running out of what they need to look after their animals. We can't just leave them."
The team has no intention of stopping.
Lubelska said: "If the fighting spreads, we will keep on going in. But yes, we will try to be careful."
On Friday, the two vans formed part of a queue of vehicles waiting to enter Ukraine that was nearly a mile long, and almost all of them kept their engines running to protect their occupants from the savage cold.
On the other side of the road, the influx of war refugees continued.
Families, although mostly without an adult male, walk wearily out of their native country, suitcases trundling behind them.
The reception area near the Medyka border crossing was its customary sea of exhausted, disorientated Ukrainians; some warming themselves around barrel fires; many joining yet more queues to continue their journeys into Poland.
But amid the sadness, there was a moment of joy. A shout of recognition, of the most profound excitement, and a man ran onto the road, looking back towards Ukraine.
From the direction of the border, a black Labrador charged to greet him.
The dog had initially been left in Ukraine and then brought to Poland by neighbours, according to a bystander.
The pair embraced, rolling around on the tarmac. It was nothing compared to the reunion of war-torn families.
Yet in that moment, it was also everything.
