Is it permissible to continue communicating with a girl with the intention of marriage, after her guardian's knowledge and consent, taking into account that the communication is free of Sharia-prohibited matters, and that it is due to the difference in countries and the impossibility of meeting?

If you are able to marry a girl, then propose to her and marry her. Otherwise, it is not permissible for you to speak with her or message her, because that is a gateway to fitnah (temptation/strife) and a means to corruption, even if the conversation is about permissible matters. The scholar Al-Khadimi said in his book Bariqat Mahmudiyyah: "Speaking with a non-mahram (definition) young woman is not permissible without need, because it is a place of fitnah."