Shasta enters Narnia in the dead of night without even realising it. He is guided to his destination by Aslan, who stops him falling off the trecherous paths which he must take. He seems unphased by his entry, as if it just had to happen at some point. And neither is he phased by meeting a talking hedgehog, despite never having encountered one before. In fact the only things that surprises him seems to be the centaurs. He is pleased with the new country, and even more pleased to find himself officially a part of it. He is relectant to take his rightful place as king, but only because it pushes Corin from the throne- he is a humble boy.
In contrast, Bree is a proud creature. She does not want to enter Narnia until she is as good as she can be. She does not want to seem less than the other horses in Narnia. She does not believe the tales that she is told, although she uses the language of Narnia. It is only when Aslan teaches her that her behaviour is foolish that she is ready to enter Narnia.
Avaris has been brought up to see Narnia as a dangerous place, and yet she is eager to get there. She wariness still remains in her attitude towards Aslan- since he is a lion, and lions are dangerous, he too must be dangerous! And yet she is ready to accept his reassurance that his paws are velvetted and will not hurt her.
Hwin has always longed to see Narnia, his home. He is not bothered how they arrive in the country, he just wants to get there. And, knowing that Aslan is the deliverer of Narnia, he puts his life in his paws (literally), despite his fears.
Now let us take a look at Rabadash. Rabadash tried to conquer Narnia, to overcome the power of its army. Of course, all that happens is that he is made to look a fool. He will not learn to be humble and instead tries to scare away his enemies, he is unable to see when he is defeated. And so he leaves Narnia under the control of Aslan and all who follow him- tame, lowly and a fool. Aslan forces peace upon him and makes him rule his own country in a much humbler manner. He will never be allowed to set foot in Narnia again!