After crossing the Polish-Ukrainian border, are non-Ukrainian passport holders allowed to travel to other EU countries and apply for refugee status there?

Under the current legislation, the answer to this question is determined by the grounds on which a non-Ukrainian foreigner has crossed the Polish-Ukrainian border. 

If said foreigner has a right to visa-free movement in the EU (a list of countries, whose foreign residents are entitled to visa-free movement can be found here:: https://eur-lex.europa.eu/legal-content/PL/ALL/?uri=CELEX:32018R1806 – Annex I), he/she will be able to apply for international protection in another EU member state. 

If the foreigner holds a residence permit or visa issued by another state, he/she will be able to apply for international protection in the state that issued the visa.  

If the foreigner holds a visa issued by any EU country, he/she will be able to travel within the EU for 90 days in every 180 days, however he/she should apply for international protection in the country that issued the visa. Otherwise, he/she may be sent back to the issuing country to have his/her application for international protection reviewed. 

On the other hand, if the foreigner in question has crossed the border on the basis of a permit issued by the Commander in Chief of the Border Guard (i.e. a residence permit valid for a period of 15 days), then according to the law he/she is only entitled to stay in the territory of Poland. If he/she crosses the border and applies for international protection in another country, he/she may be transferred back to Poland on the basis of the so-called Dublin III Regulation. Naturally, whether the transfer actually takes place will depend on the state in which the foreigner submits the application, and on whether the state completes all the formalities on time. Furthermore, there are some humanitarian or family circumstances that may result in refraining from transferring the foreigner to Poland. 

It should also be noted that individual EU member states may adopt different political decisions with regard to transferring foreigners back to Poland on the basis of the Dublin III Regulation. Other EU states, despite the right to transfer a given foreigner back to Poland, may nevertheless agree to consider the application for international protection filed by said foreigner.