All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
Now then:
For the polygamous husband, he must deal justly with his wives as much as Allah ordered him to do justice. Allah Says (what means): {…then marry those that please you of [other] women, two or three or four. But if you fear that you will not be just, then [marry only] one or those your right hand possesses. That is more suitable that you may not incline [to injustice].}[Quran 4:3]
It is confirmed in Musnad of Imam Ahmad that ‘Aa’ishah said: “The Prophet of Islam divides sharing amongst his wives and he deals justly with them and says: “O Allah! It is my dividing share; do not reproach me for what I don't own (have) as it is You Alone Who owns every thing.”
It is reported that Abu Hurayrah narrated that the Prophet said: “When a husband has two wives and does not act justly between them, he will come on the Day of Judgment with one side of his body hanging down.” [Abu Daawood, Ahmad, An-Nasaa'i and others]
From this Hadeeth we can see that if a man decides to take more than one wife, it is required that he treats both (all) of them justly. So while polygamy is permitted in Islam, it is bound by rules and set of morals prescribed according to justice and for the common good of society as a whole. As for the travel of a married man, Muslim Scholars have agreed that it is permissible for him to travel alone (without any wife) or with all of his wives.
Muslim Scholars have also addressed the issue of the married man traveling with one or less than all of his wives. However, they differ in their opinion as to how these wives are to be selected. One opinion is that the husband can select whomever he wants to accompany him.
Another opinion is that the husband selects one of his wives by casting lots among them, and he is to travel with the one(s) selected by the lot. In this way, the feelings of the wives are appeased. This method is the more preferable, as the majority of Muslim Scholars are agreed upon it and as it is mentioned by Ibn Qudaamah in Al-Mughni and in accordance with the guidance of the Prophet .
‘Aa’ishah said: "Whenever the Prophet intended to set out on a journey he cast lots amongst his wives and he took one with him in whose favour the lot was cast.” [Al-Bukhari and Muslim]
If a husband makes a journey with one of his wives for seeking knowledge or for treatment, then if it is decided after lot-casting, he does not have to divide the share (of night) to the rest of his wives after he comes back from that journey, as it was not reported from the Prophet that he did so. But if he travels with one of his wives without lot-casting, he has to cover the absent nights for the rest of his wives after he comes back from the journey. However, any wife who wants to travel of her own will to her relatives or to others with the permission of her husband, she cannot claim the missing nights, as her share is dropped due to her own choice.
To perform justice when spending the night with his wives, the husband has to stay the whole night with every one of them; he is not allowed to go to the other wives unless she permits him to go for some urgent need or for a necessity which would be valued according to its importance.
A husband has to deal justly with his wives as best as he can. It has been reported that the Prophet said: “When a husband has two wives and does not act justly between them, he will come on the Day of Judgment with one side of his body hanging down.” [Abu Daawood, Ahmad, An-Nasaa'i and others]
There are many other Ahadeeth which stress the need of justice amongst one's wives and the Prophet's guidance on that is very famous and well-known.
Based on the above, one knows the danger of injustice with wives and that is why The Almighty Allah has advised the man to confine himself to marry one woman only if he fears dealing unjustly with his wives. One has to be as just as he can with his wives and to be fearful to Almighty Allah and to seek Allah's forgiveness from what one is not capable of, just as the Prophet said: “Oh Allah, this is my (way) of sharing what I own (have); do not reproach me for what I do not own as You Alone own (every thing).” [Ahmad others]
One must know that the day belongs to the night of the wife with whom the night was spent and one is not supposed to go to his other wives except if there is need and necessity to do so, such as to inquire about her condition or the condition of his children just like the Prophet did.
For all of this, one must introspect and decide for oneself and be fearful of Almighty Allah and always be aware of His Presence as He is cognizant of all that is in the hearts.
Allah knows best.