Passport checks on the trip

It's not the EU it's the Schengen - the two are not the same.

And Serbia is a member of neither so the border control experiences you describe are as expected.

https://en.wikipedia.org/wiki/Schengen_Area#Current_members

Thanks. I meant to say Schengen, but your correction is appreciated.

All as expected:
Hungary -> Croatia: Exiting Schengen Area (Croatia is in the EU, but not in the Schengen Area)
Croatia -> Serbia: Neither in Schengen Area
Serbia -> Romania: Neither in Schengen Area (Romania is in the EU).

Just learn which countries are in Schengen and which aren't. When you first enter Schengen and when you leave Schengen your passport will be stamped. Otherwise, free travel.

For all other countries, expect immigration checks.

As others have said, it has nothing to do with being in the EU.

Human trafficking and illegal immigration are main issues for border checks.

Experience during May trip to Ireland.

Passport stamped coming into Dublin.

No stopping, nothing at all, crossing into Northern Ireland nor crossing back into the Republic.

Flew to Leeds. Passport looked at quickly and waived through, no UK stamp.

Flew back to Dublin. Passport looked at and second Irish stamp.

Flew home to US. Passport stamped in Dublin "preclearance". Annoyingly, I went through airport security then a second preclearance security.

UK and Ireland still have as soft border, no stamps. You will always have to clear out through Immigration and Passport control leaving UK or Schengen to go to the other, or from either to the US. The pre-clearance in Dublin is actually your entry into the US, allowing you to walk off the plane on landing and leave directly. There are two security checks, because the first is the standard security check to enter airside of Dublin airport. The second is because you are now in the US and subject to TSA screening, which is not the same as European screening, it comes after you have gone through Passport control..

Not to hijack the thread further...but

While the UK and Ireland have "soft borders" under the Common Travel Area, my experience was the same as Brad's in that in flying from the UK to Dublin, I was required to go through Irish Border Control, passport (US) checked and stamped, and as I recall, even the stay was limited to 3 or 4 weeks. Some are surprised by that (Oh my, you must be mistaken...no checks ever) but my experience, and Brad's above beg to differ. I have learned though it seems to be linked somehow to RyanAir and how they operate. (my flight was Leeds-Dublin as well)

As for intra-Schengen travel, even then, over the years, on occasion, border agents might board a train at one stop and check documents until the next stop.

Probably the oddest controls I saw was in 2000, prior to the Czech Republic being party to the Schengen Agreement, at the German/Czech border we only had to slow and hold up our passports, never stopping the car. On return, I recall stopping at the Czech side of the border for other reasons, but do not recall any German border controls, unless they were housed in the same facility.

I have had my passport checked when arriving on a flight from Amsterdam to Lyon and from Rome to Lyon, both times during periods of higher-than-normal security due to recent terrorist activity in France. Just because two nations are both in the Schengen area doesn't mean there will never be passport checks when traveling from one to the other -- just that there usually are none, at least under current practices.