Nisaa
No, hayd itself does not change because you are travelling. Prayer stops for the whole of hayd exactly as it does at home, and it is not made up afterward, and any fasts missed for the bleeding are made up the same way once you are pure. Travelling neither adds a restriction nor removes one. What it does raise are two practical questions that only come up because you are away from your own bathroom and your own mosque: what happens to a fast that breaks for two reasons at once, and what to do about Ghusl when there is genuinely no water to be had.
If your period starts mid-fast while you are travelling
If this happens during Ramadan, that day's fast is invalid the moment bleeding starts, exactly as it would be at home. Travelling is its own separate ground for not fasting, entirely apart from hayd. Allah says (interpretation of the meaning): “and whoever is ill or on a journey, the same number [of days which one did not observe Sawm (fasts) must be made up] from other days.” [Al-Baqarah 2:185]. Having both reasons present at once does not create a second day owed. What you make up is the same single day you would owe for the bleeding on its own, before the next Ramadan arrives.
If you become pure while travelling and cannot reach water for Ghusl
This is the situation travelling actually changes. Ghusl is owed the moment hayd ends, but a flight, a remote hike, or simply nowhere private to wash can make performing it genuinely impossible or unsafe at the time. The Qur'an addresses exactly this, in the same ayah already cited for entering a mosque in Janabah: “and if you are ill or on a journey or one of you comes from the place of relieving himself or you have contacted women and do not find water, then seek clean earth and wipe over your faces and your hands [with it]” [An-Nisa' 4:43]. This is the same concession that lets Ghusl for Janabah be replaced by tayammum when water cannot be reached, whatever caused the impurity. It is also the concession behind the well known hadith of 'A'ishah's lost necklace, when a delay searching for it during a journey left the Muslims with no water at prayer time until tayammum was revealed (narrated in Sahih al-Bukhari and Sahih Muslim). Tayammum stands in only for as long as the genuine difficulty lasts; once water is safely reachable again, Ghusl should be performed properly.
Praying once you are pure again, if you are still travelling
Purity from hayd does not end your status as a traveller, and travelling does not change any ruling of hayd. Once you are pure, you simply resume praying the way any traveller does, meaning you may shorten (qasr) the four rak'ah prayers, exactly as you would if your period had never started. Allah says: “And when you travel throughout the land, there is no blame upon you for shortening the prayer” [An-Nisa' 4:101]. Ya'la ibn Umayyah once asked 'Umar ibn al-Khattab why this concession still applied once the fear mentioned in the verse had passed, and 'Umar told him he had asked the Prophet (Salla Allahu 'Alayhi Wa Sallam) the very same question and been told it is “a charity that Allah has given you, so accept His charity” (narrated by Muslim).
If the travel in question is Hajj or Umrah specifically, the rulings are more involved than ordinary travel and are covered on their own; see what if your period starts during Hajj for how the pilgrimage itself is affected.