Sir, our Gita says that casts are not by birth, cast is decided on the basis of your karmas & gunas.
In verse 1.40 & 1.41, Arjuna says that "Varnasankar in any family will lead to the destruction of that family and The ancestors of such corrupt families fall down, because the performances for offering them food and water are entirely stopped."
So my question is what he meant by varnasankar here? my understanding of varnasankar is inter-cast but if cast are not defined by birth then it means if any person who is Brahmin by his karma get married to a girl who is vaishya by his karma then all the problems described above happens with them or the above line said by Arjuna have no meaning? Please clarify.
Arjuna is an IGNORANT PERSON. There is absolutely no need to attach any significance to what he is saying that too at the time of distress. Even in normal circumstances, without understanding the essence of Vedas - it is difficult for anyone to grasp such concepts as Caste.
So please do not try to analyse his statement. If he was talking any sense Lord Krishna would have answered his concerns. Since he was lamenting like a woman, Lord Krishna actually shouts at him to stop such nonsense in the verse no 2.2 and 2.3.
Thanks for asking this question. Hope you will not ask for further clarifications based on Arjuna's words.
Incidentally, your understanding that caste is not based on birth is correct.