Are North Cypriot land border crossings equipped with passport stamps?

Here is my explanation of how the system worked when I traveled there in 2013. Some details may have changed but the broader picture has not.

(This description makes use of the terms "North Cyprus" to refer to the predominantly Turkish speaking "Turkish Republic of Northern Cyprus" nation-state that has limited international recognition and "Republic of Cyprus" to refer to the primarily Greek speaking country which is a member of the EU which makes claim to the whole island but only controls the southern portion.)

1. When your plane lands at Ercan Airport you are landing on territory controlled by North Cyprus. Since North Cyprus considers itself a sovereign state, you will pass through that country's immigration control shortly after landing. The border authority generally issues a stamp to all arriving non-Cypriots (i.e. to both EU and non-EU citizens alike.) By default they will stamp your passport although you can request that they stamp an official piece of paper (provided by the immigration agent) if you do not wish to fill your passport with needless stamps or if you are concerned about possible future issues with the Republic of Cyprus immigration authority.
2. When crossing by land from North Cyprus to the Republic of Cyprus you will first pass through a North Cyprus Emigration Control since North Cyprus considers you to be leaving their country. The agent there will place an exit stamp on the same medium containing your North Cyprus entrance stamp.
3. You will then walk through the demilitarized zone.
4. On the other side, you will arrive at a checkpoint controlled by the Republic of Cyprus. At one point in time, this checkpoint may have issued passport stamps however, this is the station which other internet accounts may report as no longer having passport stamps. This is because the Republic of Cyprus considers the entire island to be part of this country and thus at this point, you are not entering the country but rather passing from the "non-government controlled zone" to the "government controlled zone". The letter of the law states that non-EU citizens who not arrived on Cyprus at a government controlled port (such as an airport in the Republic of Cyprus) can not pass this point however this is only occasionally enforced. (I, a non-EU citizen, was able to pass without issue.) However, should a non-EU citizen then try to depart Cyprus from an airport in the Republic of Cyprus controlled zone, they are more likely to run into difficulties. As Britain is part of the EU (effective 2017) you do not need to worry about this caveat. (I believe that this caveat is because EU citizens do not need to keep records of their entry to any EU country or some similar rule.) You will likely leave this checkpoint being unable to obtain a stamp.
5. a. If you choose to then travel back into North Cyprus from the Republic of Cyprus, there is no emigration or other checkpoint before entering the demilitarized zone. Once through the demilitarized zone, you will encounter a North Cyprus immigration control. This control is similar to the one at Ercan Airport in that it will issue a stamp however, by default they will stamp a piece of paper unless you ask them to stamp your passport. If you then depart from Ercan airport, you will go through an emigration control at the airport which will issue an exit stamp.
   b. If you depart by air from an airport in the Republic of Cyprus controlled zone, you will pass through an emigration control at the airport. This emigration control will expect to match their exit stamp with their entry stamp. Since passport holders of EU countries are generally not issued stamps from other EU countries this step will be skipped and you will be allowed to pass regardless of entry circumstances. Non-EU passport holders who try to exit without a corresponding entrance stamp are likely to run into problems at this point where the punishment can vary from being yelled at by the border guard to theoretically being deported to the country which issued your passport.

As an aside, even if a border guard wouldn't normally stamp your passport you can politely ask for a stamp and they will normally provide one.

I crossed that border a year ago in Nicosia. All I had to do is to show the passport. Not sure if they even have the stamps, as the north Cyprus is not admitted as an independent state by most of the countries (afaik, all except Turkey). You may ask for that, however, it is unlikely, that such things exist.

We crossed at Astromeritis/Zodhia on January 6, 2018, and the TRNC side is most definitely equipped with stamps. It was clear the officer hadn't used it for a while, as she had to first get it out of a drawer and then change the date, month and year of entry, but she was happy to stamp our passports.