The person designated the household head (i.e., the person with RELATE = 101) does not change between survey panels unless that person moves out of the household. An adult child moving in with a parent will not replace their parent as the household head. More broadly, the Census Bureau discontinued the use of the term “head of household” beginning with the 1980 CPS and replaced it with the concept of a “householder”. A householder is a person in whose name the housing unit is owned or rented. The householder is determined when the household enters the panel; they are listed first in their household roster (assigned PERNUM = 1) and are designated the “reference person” to whom the relationship of all other household members is recorded (see the Census Bureau’s subject definitions glossary).
My recommendation is to begin by obtaining the month that each person first appears in the CPS panel. With your dataset that includes all monthly samples from 2021-2023, you can group matching CPSIDV values and sort them by YEAR, MONTH. Generate a new variable for each group of matching CPSIDV values that equals the first YEAR, MONTH combination that they are observed in. You can then compare this value to the first six digits of that person’s CPSIDV value (which is the same as their household’s CPSID value); these first six digits (or entirety of the variable if using CPSID) index the year and month that the household entered the CPS. Therefore, you will be looking for cases where the date the household entered the panel does not match the date of the first appearance of the person’s CPSIDV value. For example, an individual that first appears in the February 2022 sample in a household with a CPSID value that begins with “202201” would have moved in between January and February 2022 (i.e., they became a usual resident; see table 3-2.4 in technical paper 77 for how household membership is determined). Note that you will only be able to identify these movers for households whose first month-in-sample is within the years included in your data (i.e., MISH = 1 in your earliest observation period). If your sample begins in January 2021, you will not have the full history of households who entered before this date.
The forum post that you link to that suggests checking the first 6 digits of CPSID vs CPSIDP is inaccurate; the first digits of CPSID and CPSIDP/V will match regardless of when the individual first appears in the sample. Thank you for bringing our attention to this! We will revise our team’s response to that post to ensure it doesn’t cause confusion.