Can Muslims visit shrines and tombs while traveling?

Visiting graves is encouraged

The Prophet ﷺ said he had once forbidden visiting graves, but now people should visit them, because it reminds them of death (Sahih Muslim 976, 977). He taught a greeting for the dead: "Peace be upon you, people of these dwellings, believers and Muslims..." and a dua for them (Sahih Muslim 975). See What do I say when I pass a graveyard?

So stopping at the grave of a famous scholar in Istanbul, a Companion's tomb in Damascus, or a royal mausoleum in Morocco can be a meaningful reminder.

The line that must not be crossed

All worship and every request belong to Allah alone:

Near his death, the Prophet ﷺ warned against those who turned the graves of their prophets into places of worship (Sahih al-Bukhari 1330). At some shrines you may see people:

• asking the saint for a child, a cure or a job,
• bowing or prostrating toward the tomb,
• tying cloths, throwing coins or circling the grave like the Kaaba.

Scholars agree that calling on the dead instead of Allah is shirk, and most say the other practices are serious innovations to avoid. Make your dua to Allah directly; you can pray for the person buried there, not to them. See What is intercession in Islam?

Traveling specially for a tomb

The Prophet ﷺ said not to set out on a special journey except to three mosques: the Sacred Mosque in Mecca, his mosque in Medina, and al-Aqsa (Sahih al-Bukhari 1189). Scholars differ on what this covers. Many say it means you shouldn't plan a trip just for a shrine; others say it's only about mosques. Either way, visiting a grave while you're already in a city for another reason is fine.

Shrines of other faiths

Tombs of Christian saints or Hindu holy men are often tourist sites. Visiting as a respectful observer is similar to visiting other religious buildings, without joining in rituals. See Can Muslims visit a Hindu or Buddhist temple?