Obviously everyone is different and without knowing the exact situation, all is only speculation. But in my opinion, the first problem is a man offering a woman to come to him. I think this can be a sign of one who is not serious. My feeling is the man must make the effort to go to his woman and see her in her own environment. I think that if you are wanting to combine the two cultures that a man would want to see where his woman is from, learn about her life in person. and also to meet her family and friends. You can learn a lot about a person in their home environment. Not so much on a vacation. Can have a good time with almost anyone on a vacation for a week. Much easier to invite women to himself and see who arrives. Very not serious. I would not consider visiting a lady until I felt that the relationship was more serious. It is too large of a step.
The men being quiet? That is a way to avoid confromtation. It is also extremely disrespectful and I would never do this with a woman that I had any interest in. When I met my last girlfriend two years ago, I made a point to explain the situation and why I was leaving the site to all that I had even a friendly relationship during the correspondence. It is simply the right thing to do.
I think that better you learned that this man is not a serious, respectful man and he is either afraid to make the commitment or was just playing games. Better to run from this sort of man anyway. You can do better Larisa. There are too many wimpy, immature, disrespectful men these days and this is just the example of it.