Based on this Tumblr post:

bucksterbarnes:

Did the Howling Commands have a funeral for Bucky
Did they bury a box with some smokes and a bottle of whiskey
Did Steve tell themyou guys go ahead, I'll stay here for a while
Did he talk to the fresh churned dirt and tell itcan't believe you're gone, you big jerk - can't believe you left me - I know it was my fault, and I'm sorry, but I still can't believe you're gone
Did he stay there til it got dark and cold and Peggy came out and brought him a coat
Did she decide not to talk to him about it then - not til he was ready, drinking alone in some wreckage of a bar
Is there still some stone on an old battlefield in Europe with Bucky's name scratched onto it, marking some decades-old alcohol and decaying cigarettes and a memory of a guy who's alive, but doesn't really exist anymore

On a mountainside in Austria, there is a small flat stone with a fading scratched out scrawl across the surface that reads: B. Barnes. 1917-1944. He could still find it if he went back, he's sure. The coordinates are burned into his brain like a brand. There's little about that day that has faded even the slightest bit from memory.

Steve thinks about it as the plane takes off, as he clutches numbly at the coarse fabric strapping that keeps him on his feet. They're all thinking about it really… it's been less than an hour since they placed it there.

They had all stood unbelieving at the mouth of the ravine, grim and silent. Dugan looked like someone had punched him straight in the stomach, and Captain Rogers made an awful choking noise now and again that the others did their best not to notice. Jones was the first to speak.

"We've gotta do something for him." He'd said softly, shoving cold hands deeper into his pockets as he glanced sidelong down the line. "Sarge deserved somethin' better than this…"

"Like what?" Morita had asked, a little despondent. "We ain't gonna find him in that mess…" He tilted his head toward the snowy jagged crevice in the earth that sprawled wide before them.

"If we didn't find anything the first time, we'd certainly never do it now." Falsworth joined them, trying his best to sound matter-of-fact. "But I agree. Something must be done."

Without a word, Dernier produced a small flask of whiskey he'd sweet-talked out of a local on their way through France, and Dugan produced a pack of smokes. They'd been Bucky's favorite brand, and Dum Dum always tried to keep an extra pack around in case 'Jimmy' wanted to bum one.
Steve tugged out an old photo of his friend that he'd carried around with him since he joined up. It was a beat-up grainy thing, half blurred with age already, but Barnes' 100-watt smile was clear enough, even if that Barnes had died long before the one they'd just lost. None of them commented on this as the photograph joined the liquor and cigarettes.
It's not much, they all realized, and it's nowhere near enough, but it's all they had. It'll have to do.

Steve laid the first load of dark cold earth over the little pile in silence, eyes dry and red. His back was straight as a steel rod, and his jaw was clenched tight and rigid. Dum Dum stepped up next, and they each cycled through laying their little bit of soil in place. Steve finished it with a stone they'd found, hastily etched while the others dug.

He stood and stared at their handiwork, eyes distant, until Morita stepped up and set a hand on his arm. Steve startled and glanced at him, smiling wanly.
"We should pack up camp and get to the extraction point…" The Captain sighed, looking decades older than his meager 25 years. He started to turn away.

"We'll get it, Cap." Morita told him kindly, the others already moving. "Don't worry about it."
Dum Dum paused for a moment as he passed and clapped one enormous hand against Steve's shoulder. Then he was gone again, shoulders slumped, but steps steady. They had work to do.

Steve had stood a long time in silence until the others were nothing but distant background noise, moving efficiently around the base camp. Where he stood there was nothing else to hear but the empty whistle of the wind, nothing to see but the dark damp earth, chilly against frigid snow, staring back at him.

"I can't believe you're gone, you big jerk." Steve murmured, crouching down out of the wind, as if Bucky was sitting on top of the stone, listening. "Can't believe you left me…" He had to stop and clear his throat or he'd break down and never get the words out. "I know it was my fault, and I'm sorry… But I still can't believe you're gone."

He reached out one gloved hand and brushed a few errant flakes of snow off of the surface of the stone. Damp trails followed his fingers, like tears, down the hard, rough rock. The soil was already turning frosted-white under a dusting of fresh flakes. The wind howled mournfully in the canyon below.

"You always did have to be first at everything, huh?" He said softly, voice catching. "Never could fuckin' wait for a guy to catch up…" He reached under his uniform and tugged out a scuffed ball-chain with two little metal tags attached, coiled them in his hand and pressed them down into the soil just below the stone. "Hang onto those for me, yeah pal? No point carryin' em around now… Next'a kin is right here."

He stayed silent a long time after that, just watching the snow slowly bury all that he had left of his best friend.
When the mound was solidly white and the stone barely visible, Agent Carter appeared behind him from the transport plane, a thick wool coat -the biggest they could find- wrapped over one arm.

"Captain Rogers, your men informed me that I could find you here." She'd said, a gentle edge to her usual crisp tone. "It's time to go."

Her eyes flickered to the lump beneath the snow, but she didn't ask. Just held out the coat to him then guided him back to the plane in silence, one hand lightly placed on the small of the big man's back.

It's ironic, Steve thinks to himself, as he stares down at Bucky's - the Winter Soldier's- file… that he could still go back and mourn his best friend if he wanted to. He could stand right where he was the day that Bucky fell, and he could say goodbye all over again. Because this man he's found? The one he's desperately trying to track down? He might be James Buchanan Barnes. He might still be Steve's best friend, and the one man on earth he can never, ever give up on… but he's not Bucky anymore. Bucky died in 1944.
He's not sure who he's looking for now.

Wind whistles cold and stinging through lank unkempt brown hair, tangling it in his eyes. The Soldier makes no move to push it back. His hands, flesh and metal, are busy. The file he found said it was here. It was right here.

He finds the stone first. Flat and worn and caked in frost. Just faintly, he can still make out a 'B' and what looks like '44'.
There's a slight lump beneath the snow just below the stone, and he digs up his own grave with shaking hands, blood seeping from the cracked skin of his fingertips, the ragged, tattered nails of the flesh hand. He ignores the blood. It's hardly new, and it doesn't even register as pain; not after what was done to him over and over again.

A small grimey metal flask comes into view at length, and the mouldering remains of red-tinged paper scrap and rotted vegetation under that. Tobacco. They used to be cigarettes, his mind supplies, unasked. My favorites… it elaborates. He shakes the thoughts away. There's an unintelligible scrap of thick paper in there as well, covered in a grey-black blur, though he isn't sure what that is. A photo maybe? He ignores it in favor of the glint of metal that has caught his eye in the discarded dirt. A set of tags on a chain, filthy and faded with age, come free when he yanks on them. The brittle chain disintegrates, but the tags land in the snow beside his knee, and he picks them up cautiously.

Rogers, Steven G
011 546733 SSR - H C
James Buchanan Barnes
43B 212 S. Moore Row
New York, NY M

The pressed letters are all but gone on one of the tags, but that one must have sheltered the other, because the second is clearly legible. His own name stares up at him from the filthy corroded metal.
Next of kin his mind volunteers again. He was… The Soldier pockets the tag and the flask, pushing the earth back in over the rest. He stands on suddenly unsteady knees, planning his next move.
The online files, the museum, the archives he broke into… it all led him here, and now he has proof, small and metal and ancient, riding in his pocket, that Rogers told him the truth. What he is, what he's suffered… he knows who is responsible, and what they made him do. And now he's going to go return the favor.

HYDRA is engulfed in flames, former SHIELD stomping out the ashes and dealing with the stragglers, when Steve opens his front door to find Bucky standing there, nervously flexing his metal hand open and closed, open and closed.

"... Bucky?" He asks, because he can't help himself.

"No." The metal-armed man replies, finally raising his eyes off of the welcome mat. "But I'm the next best thing."