When the CdP expires during the journey or the 25 pages are full you will need a follow-up carnet for your onward journey and apply with the same documents as for the initial application. You send the application again to the issuer a few weeks before it expires (please check with the issuer in charge of your home country how far in advance they need everything). Depending on the travel region, the deposit may also have to be increased. Usually, the issuer can also send the new carnet by courier to many countries around the world against reimbursement of the costs.
With the old CdP still valid, you then leave the current country, have it stamped out and enter the next country with the new carnet. In some countries it is also possible to have the new carnet stamped within the country. It is best to find out about this locally. Send the old carnet back to the issuer in the safest possible way. Please note that some issuer demand a copy of the first inside page of the new carnet, which contains the new stamps and the vehicle data.
Another option when the CdP expires is to extend it in the country where you are at that time. The extension is usually possible for up to three months but does not entitle you to travel on to another country. You must apply for this extension at the issuer and the local automobile club in the country (so it works only in countries where there is a FIA or AIT club) of travel about four to six weeks before the carnet expires. In some countries, special rules apply to the extension. In Australia, for example, we need the approval of the Australian customs authorities. The decision of the customs authority depends, among other things, on the period of validity of the personal visa. Just one example.
Handling of the CdP always sounds terribly complicated, but it's not. There are so many travelers on the road with it, you just have to familiarize yourself with it once. By the way, since North America and South America were also mentioned in the question, please check exactly where you really need the CdP an where not. We have a lot more information about the CdP in the
OB boot camp article abot the CdP.