Military units absorbing stragglers from other nations?

Is this set in a real, historical period - if so, which one, and which culture? Or is it fantasy? Or is it contemporary? Even if it's contemporary, what countries are they from? Different countries will have different rules and procedures for their armed forces. If it's fantasy, you can do whatever fits your worldbuilding and story. And even for real world settings, people don't always follow the standard protocol and may do what best suits the situation.

To give a very general answer, I'd find it plausible if they joined forces and used the expertise of the soldiers they rescued, rather than just having them tag along.

No actual military expertise here, but based on what I've heard/read: if the rescued and the rescuers come under attack, it would be all hands to the guns. The rescuers may delegate someone to keep an eye on them, but the more fire-power the better.
If they're all just travelling from point A to point B, probably better to just treat them as 'passengers', keep them safe, keep them fed, keep them moving - that way there's no need to deal with conflicting command structures. The person in charge of the survivors may have trouble deferring to the leader of the rescuing group. Or, the troops may just get confused as to whose orders to follow.

Also, the survivors may be wounded or upset by what's happened to them. The rescuers may not want to add to their stress level.

I would say that allies would quickly jump into bed together, so to speak. A lot of rules go out the window in war, and although I can't think up any historical examples off the top of my head (I'll see if I can't find any; I suspect that it happened in WWII, if not more recently), I can see the two units joining together on a temporary basis. How integrated they are would depend on how much leadership is left from the stragglers' unit. If they've got decent leadership, they'll probably come together to form something of a sub unit (think a platoon, which is part of a company, which is part of a battalion) under the bigger unit/rescuers. Without leadership, they might just integrate into an existing part of the unit.

In my time in the Navy, we worked with a lot of countries that our nation is close to, and our policies and procedures are close enough that working together is ridiculously easy. I can honestly see this happening in a heartbeat.

I never served in the military, but my father was a career officer and I have many other family members who have also served. I also grew up on military bases and now live just a few miles from one. But I am not now nor have I ever been in the military.

That said and having been around many combat troops, I would think they would be careful around stragglers who show up claiming to be allied troops without properly vetting them. Seems like it would be a great way of infiltrating the other side to me. I'd want more than a uniform, some paperwork and some guy's word before I trusted him.

Just my opinion, though

If you want a great example of this, grab a copy of the movie "Sahara" (the one with Humphrey Bogart). It starts off with an American tank that winds up picking up British soldiers at a bombed-out hospital and then adds a Free French soldier, a Sudanese soldier with his Italian prisoner, and eventually, a German pilot (who is a pain the rest of the movie). Sure, the actual events are fictional, but there were actual military personnel involved in writing the story, so it gets a lot of stuff right. There were also examples of British/Canadian or British/US units working together in the early days of the Normandy invasion as units got separated and ran into each other, esp. the night before the invasion (British/US paratroopers primarily). Also, US units got co-mingled (82nd and 101st paratroopers formed ad-hoc units until they could find their parent units).

However, you also potentially run into the issue that we had at the Battle of the Java Sea (1942). There was a combined Dutch, Australian, British, and US fleet going up against the Japanese, but no one in the US fleet spoke Dutch, so all the orders had to go from the Dutch commander through the British 2nd-in-command and then to the US fleet. Once the battle started, communications were garbled at best and the combined fleet was cut up piecemeal because they couldn't coordinate well.

This has happened in the past, and likely will in the future. Allies tossed together by circumstances will defer to highest rank, whatever military that is. Happens a lot less frequently now days because command and control is a lot better, as is communications. Plus, it's pretty easy to get a rescued group back to their own military now.

Jeff

WW2 some military folks of countries defeated by Germany turned up in the UK and continued the fight. See this for example - https://en.wikipedia.org/wiki/Polish_Air_Forces_in_France_and_Great_Britain
Examples of both Poles in RAF units and Polish Airforce units.
Have a tag in my head that there may also have been ships that came over. Not googled that, but something else to look for.