JOHN YANG: For months, Ukraine's military has been grinding through a counteroffensive designed to liberate Russian-occupied territory.

Kyiv has admitted that it's been slow going.

But, over the last 72 hours, Ukrainian soldiers have made what the White House called today notable progress.

Nick Schifrin reports on the counteroffensive and a population eager for that progress.

NICK SCHIFRIN: In a city at war, the first day of school is held underground.

(SINGING) NICK SCHIFRIN: In a Kharkiv-metro-station-turned-bomb-shelter, children sing the words of their national anthem: "Our enemies will die and well live happily in our land."

(APPLAUSE) IHOR TEREKHOV, Mayor of Kharkiv, Ukraine (through translator): After our victory, we will celebrate the first day of school very differently.

(GUNSHOTS) NICK SCHIFRIN: But that victory has felt a long way off.

For weeks, Ukrainian special forces in Southern Ukraine have been fighting through thick Russian defenses.

They have taken casualties, targeted on the ground by Russian artillery and from the sky by kamikaze drones.

Ukrainian soldiers have been stymied by massive mine fields pockmarked with craters.

They use American technology and armored vehicles to hunt and detonate mines, trying to create a corridor to drive through.

And, last week, they achieved, a breakthrough a village destroyed, but recaptured.

Robotyne on Ukraine's most critical access toward the city of Melitopol on the way to Russian-occupied Crimea.

And, in the last two days, Ukraine says it also captured Verbove.

Both villages lie within heavy Russian fortifications marked in red.

In certain places, Ukrainian soldiers have now breached the first line of defenses and are confronting the second line, Deputy Defense Minister Hanna Maliar said on Ukrainian TV today: HANNA MALIAR, Ukrainian Deputy Defense Minister (through translator): The enemy is much more fortified there.

And, in addition to the mining, we also see concrete fortifications.

And our armed forces have to overcome a lot of obstacles in order to move forward.

NICK SCHIFRIN: Ukraine also appears to be moving forward with attacks inside Russia.

On Wednesday, drones struck a Russian air base.

Satellite imagery showed military planes in Pskov before and after the attack, some destroyed.

The base is only 20 miles east of NATO member Estonia.

It's also 400 miles from the Ukrainian border, which just so happens to be the range of what Ukraine says is a new weapon launched this week.

But the attacks on Russia are launched from Russia, senior adviser Mykhailo Podolyak said today.

MYKHAILO PODOLYAK, Adviser to Ukrainian President Volodymyr Zelenskyy (through translator): All this is being done by partisan movements or an agent network operating on the territory of the Russian Federation.

NICK SCHIFRIN: Podolyak also said today Ukraine would only stop fighting when it -- quote -- "destroys" Russia's military, even if that -- quote -- "continues for some time."

For more on the state of the counteroffensive, we turn to Doug Lute, retired lieutenant Army general and former U.S. ambassador to NATO.

Doug Lute, thanks very much.

Welcome back to the "NewsHour."

How significant is it, as we heard from the deputy defense minister there, that Ukrainian forces have breached some of the first lines of Russian defenses, most notably, some of these minefields in Southern Ukraine?

LT. GEN. DOUGLAS LUTE (RET.

), Former U.S.

Ambassador to NATO: Well, Nick, I think it's significant, because they have actually beaten the Russians on this first line of defense.

But I also think this first report should be cautioned, because there are subsequent lines at least as bad as the one that -- as severe as the one they just penetrated that lay in front of the Ukrainian advance.

So it's good news.

But it's mostly so far a tactical success, without great operational significance.

NICK SCHIFRIN: So they have gone through some of these first initial lines, which are mostly minefields, and then they come to the second line, which are concrete barriers.

And we have video of some of these barriers.

They're known as dragon's teeth.

What is the challenge there?

And you see them right there.

And there are hundreds of miles of these across Southern and Eastern Ukraine.

What is the challenge that those fortifications pose that's different from the challenge posed by these big minefields?

LT. GEN. DOUGLAS LUTE: Well, if the Russians followed their own doctrine, most of the heaviest minefields would be those in the first line of defense, so the line that reportedly the Ukrainians have already penetrated.

The subsequent lines of defense will feature, if doctrine proves true, fewer mines, but physical barriers, like the ones that you just mentioned, and also ditches, which impede the movement of both wheeled vehicles and armored vehicles.

But, there, those together still form an effective line of defense and can channel and focus the Ukrainian offensive, making it a target for a Russian counteroffensive.

NICK SCHIFRIN: Give us some perspective.

How difficult would this task be, again, across hundreds of miles, through minefields, through concrete fortifications, how difficult would that be for the American military?

LT. GEN. DOUGLAS LUTE: Look, in the menu of tactical tasks that American armored and mechanized infantry units train towards, this is the most difficult.

At the National Training Center in the Mojave Desert in California, where we train our mechanized forces, when American units attempt this task, they typically fail perhaps nine out of 10 times and have to do it over and over again to attain proficiency.

So, this is very tough.

It's especially tough for the Ukrainians, because they don't have all the assets they need.

They have insufficient mind-clearing and obstacle-clearing materials, typically manned by combat engineers.

And they also don't enjoy much air support, because the Russians still have heavy air defenses, and, actually, along the line of contact, dominate the air scene.

So, they're playing with some very significant disadvantages.

NICK SCHIFRIN: The National Security Council communications director, John Kirby, said today that there would be -- quote -- "reinforcing and pushing back on some of Ukraine's progress on the Russian side."

So what might the Russian response to some of these tactical successes be?

LT. GEN. DOUGLAS LUTE: Right.

Well, so, as we have already discussed, the good news is, apparently, the Ukrainians have penetrated the first belt.

The bad news is, there are subsequent belts, of course, as we have discussed, but, also, it's now clear where the Ukrainian focus is, which means that, in turn, in response, the Russians -- the Russians can focus their replies.

And what I would expect is attempts by Russia to attack the flanks of this Ukrainian penetration, where, arguably, the Ukrainians will be vulnerable.

So it will be a real move and a real priority afoot by the Ukrainians to deepen the penetration, but also to widen it and to protect its flanks.

And that's what we should watch for in the next several days.

NICK SCHIFRIN: And, Doug Lute, in the 30 seconds or so that we have left, we have seen a Ukrainian claim of a missile with more than 400 miles' range.

We have also seen Ukrainian drone strikes deep inside Russia.

Again, quickly, what significance do those two announcements make?

LT. GEN. DOUGLAS LUTE: Well, I think twofold.

First of all, Nick, it shows that the Ukrainians have an indigenous, homemade capability to strike the Russian homeland, and, in particular, to strike those military targets from which launches -- from which the Russians have been launching strikes on Ukraine, to include Ukrainian civilians.

So that's important.

And then, second of all, psychologically, it's a big reminder to the Russian people that they're at war.

NICK SCHIFRIN: Doug Lute, thank you very much.