Where were the Men of Lake Town during the battles of Minas Tirith and Middle-earth?

Question

Where were the Men (or rather people) of Lake Town during the Battle of Minas Tirith and Battle for Middle-earth?

Had the Men of Lake Town merged with one of the kingdoms of Rohan or Gondor by then?

Answer 1

They were busy fighting their own battles

At the same time as the great armies besieged Minas Tirith a host of the allies of Sauron that had long threatened the borders of King Brand crossed the River Carnen, and Brand was driven back to Dale. There he had the aid of the Dwarves of Erebor; and there was a great battle at the Mountain's feet. It lasted three days, but in the end both King Brand and King Dáin Ironfoot were slain, and the Easterlings had the victory. But they could not take the Gate, and many, both Dwarves and Men, took refuge in Erebor, and there withstood a siege.
_{The Lord of the Rings, Appendix B, The Tale of Years: The Third Age}

And it was only after the fall of Barad-Dûr that the seige was lifted:

After the fall of the Dark Tower and the passing of Sauron the Shadow was lifted from the hearts of all who opposed him, but fear and despair fell upon his servants and allies...

When news came of the great victories in the South, then Sauron's northern army was filled with dismay; and the besieged came forth and routed them...
_ibid.

The new kings sent emissaries to the coronation of King Elessar and had friendship with Gondor:

[The] remnant fled into the East and troubled Dale no more. Then Bard II, Brand's son, became King in Dale, and Thorin III Stonehelm, Dáin's son, became King under the Mountain. They sent their ambassadors to the crowning of King Elessar; and their realms remained ever after... in friendship with Gondor; and they were under the crown and protection of the King of the West.
_ibid.

In some other writings in the Unfinished Tales, Gandalf speaks highly of the Men of Dale and Dwarves of the Mountain, acknowledging the defeat of Sauron having started thanks to a chance meeting between Gandalf and Thorin. The context of the extract is Gandalf speaking to Frodo, the other Hobbits and Gimli in a house in Minas Tirith after the coronation of King Elessar:

“It might all have gone very differently indeed. 'The main attack was diverted southwards, it is true; and yet even so with his farstretched right hand Sauron could have done terrible harm in the North, while he defended Gondor, if King Brand and King Dáin had not stood in his path. When you think of the great Battle of Pelennor, do not forget the Battle of Dale. Think of what might have been. Dragon-fire and savage swords in Eriador! There might be no Queen in Gondor. We might now only hope to return from the victory here to ruin and ash. But that has been averted - because I met Thorin Oakenshield one evening on the edge of spring not far from Bree. A chance-meeting, as we say in Middle-earth." _{Unfinished Tales, Part III: The Third Age, III: The Quest of Erebor}

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A note on Lake-town

Lake-town was destroyed by Smaug the Dragon, after which the Men returned to Dale and rebuilt there. It was only around two years later that Esgaroth was rebuilt. Bard, king of those Men, however returned to Dale and lead from there.

Bard had rebuilt the town in Dale and men had gathered to him from the Lake and from South and West, and all the valley had become tilled again and rich, and the desolation was now filled with birds and blossoms in spring and fruit and feasting in autumn.
_{The Hobbit, Chapter 19: The Last Stage}

Answer 2

The realm of King Brand had been under threat by Sauron's armies, and around the time Minas Tirith was besieged, those armies attacked.

At the same time as the great armies besieged Minas Tirith a host of the allies of Sauron that had long threatened the borders of King Brand crossed the River Carnen, and Brand was driven back to Dale. There he had the aid of the Dwarves of Erebor; and there was a great battle at the Mountain's feet. It lasted three days, but in the end both King Brand and King Dáin Ironfoot were slain, and the Easterlings had the victory. But they could not take the Gate, and many, both Dwarves and Men, took refuge in Erebor, and there withstood a siege.

When news came of the great victories in the South, then Sauron's northern army was filled with dismay; and the besieged came forth and routed them, and the remnant fled into the East and troubled Dale no more. Then Bard II, Brand's son, became King in Dale, and Thorin III Stonehelm, Dáin's son, became King under the Mountain. They sent their ambassadors to the crowning of King Elessar; and their realms remained ever after, as long as they lasted, in friendship with Gondor; and they were under the crown and protection of the King of the West.
(Lord of the Rings, Appendix B)

This is also why the Elves of Mirkwood were not present, or the Elves of Lothlorien - they were similarly under attack.

As an aside, since it may not be clear, Esgaroth (or Lake-Town) was compassed within realm of King Brand of Dale.

Nowhere are there any men so friendly to us as the Men of Dale. They are good folk, the Bardings. The grandson of Bard the Bowman rules them, Brand son of Bain son of Bard. He is a strong king, and his realm now reaches far south and east of Esgaroth.’
(Lord of the Rings, Fellowship of the Ring, Many Meetings)