Thank you for taking the time to answering my questions.
My kids were making pizza, they put the plate on a milchig plate. Before eating they poured extra ketchup on the plate. By this time I assume that the pizza was not yad soledes bo. We keep separate containers for milchig and fleishig ketchup because of nitzuk chibur. The extra ketchup they squeezed on the plate was from the fleishig ketchup bottle (this was nitzuk chibur to hot dogs at some point in the past).
What is the status of the milchig plate?
What is the status of the pizza (this is theoretical because they already ate it) but I would like to know for the future? Would it make a difference if the pizza was still yad soledes bo? if they poured the ketchup directly onto the pizza?
You can be lenient concerning both the plate and the ketchup.
The Mishnah (Machshirin 5:10) mentions the concept of nitzok concerning matters of tumah and tahara, and the Gemara makes particular note of the concept concerning the prohibition of yayin nesech (and stam yeinam) — see Avodah Zarah 72.
Concerning regular laws of kashrus, the Rema (Yoreh De’ah 1-5:3) rules that one should preferably be stringent for matters of nitzok, but that bedieved one can be lenient.
The reason for this is that many authorities write that the principle of nitzok, whereby a flow of a liquid creates a “connection” between the source and the target, does not apply to general matters of kashrus, and is reserved for issues of tumah and for wine (see Yam Shel Shlomo, Chulin Chap. 7, no. 37 (at the end), citing from the Maharil and the Terumas Ha-Deshen; see also Biur Ha-Gra, Yoreh De’ah 105:3).
In addition, several commentaries explain that the reason for the prohibition is the connection formed by the flow in conjunction with the steam that rises from the lower food or container to the upper one (see commentaries to the Mishnah of the Rambam and others; see Taz 105:6; Shach 105:11).
In the case of the pizza this will not apparently apply, certainly in view of the fact that the pizza had already cooled somewhat.
Therefore, bedieved one can certainly be lenient.