Well they did expand quite a bit. But they were under continuous warfare against Mongols and Crusaders and each other. And ironically, some scholars (Humphreys & Marsot come to mind), have argued that they consisted mainly of refugees and slaves who were captured as part of the Mongol expansions.

The other main reason is due to their numbers. They were a tiny elite, nowhere near the numbers you would need for horde warfare. For example, Baybars al-Bunduqdarihad between 12,000 - 16,000 Mamluks in his army. On the other hand, Sultan Sha'ban is said to have had only 200 Royal Mamluks, and Sultan Jaqmaq 4,000. It varied a lot by how powerful and wealthy the ruler was.

The Mamluks formed the elite permanent soldiers. Trained from youth in Arabic, Islam, horsemanship, and fighting. They only numbered a few thousand at any one time. At times of war the bulk of the army was expanded by cheap Arab auxiliary troops, and depending on the period, Turkic and Kurdish mercenaries. Both the Ayyubids and Mamluks 'banned' Arabic speakers from being professional soldiers, although it happens anyway later on. Both the Mamluks and Ayyubids frequently also used Arab Bedouin brigades.

The army stationed in Egypt was known as al-'askar al-misri or al-'asakir al-misriyya.

The whole force in Palestine Syria and Lebanon was known as al-'asakir ash-shamiyya. And there were subdivisions such as al-'asakir at-trabulusiya, al-hamawya, as-safadya, etc.

Of the Mamluk army, there were 3 types:

1. Mustakhdamun

2. Mamlukun / mushtarawat

3. Al-Halqa

The Sultan's Mamluks (called Al-Mamalik as-Sultanya) could only be from the first 2 groups.

There were many many further subdivisions based on how the Mamluk came to serve in the army and what their roles were.

The Halqa group were free men. Non-Mamluks. Although there was within the halqa a special unit composed of the sons of the amirs and sons of the mamluks, called awlad an-nas. The Halqa began as an elite Mamluk force of a few hundred soldiers during the Ayyubid era, and then in the Mamluk era it becomes the lowest and largest part of the standing army and takes in Arabs as well.

The main fighting force was the Royal Mamluks of the Sultan. They were the most highly trained and they owned large private estates. At their peak during the Bahri period they are said to number 10,000. And these were all based in Cairo, nowhere else except for Makkah. We have a description of the army from the reign of an-Nasir Muhammad bin Qalawun, and it lists 2,000 Royal Mamluks. When a new Sultan took over he would bring in his own royal Mamluks and the old Sultan's Mamluks would be part of a different ta'ifa (faction). But sometimes the Sultans changed so fast that they had multiple tawa'if (factions) of Royal Mamluks in their service. On some occasions different tawa'if of Royal Mamluks went to war against each other. Many of these Royal Mamluks also weren't soldiers but flag bearers, masters of the stables, Arbab wadha'if (office holders), etc. It gets very complex.

During the time of Baybars al-Bunduqdari it is said that the total number of Mamluks was either 12,000 or 16,000, with Baybars' Mamluks numbering 4000. The total size of the army was 40,000.

The Ayyubid Sultan al-Kamil Muhammad and his son as-Salih Ayyub are said to have had 10,000 Mamluk soldiers in their armies. Obviously their wars with the crusaders involved a lot more than 10,000 soldiers.

Regarding Sultan Qalawun, "The historian Baybars al-Mansuri, Qalaun's devoted mamluk, and the most important authority on the kingdom during that sultan's time, states that the number of Qalaun's mamluks, comprising all regiments and ranks, was at the end of his rule only over 6,000, a number which he considers to be very high."

The main differences between the Ayyubids and Mamluks are:

1. The Ayyubids only used Turkic slave soldiers as auxiliary troops. The bulk of their armies were locals and paid Turkmen, Kurds, Arab Bedouins. The Mamluks made Mamluk soldiers the foundation of their military. Both Ayyubids and Mamluks banned Arabic speakers from being anything more than low paid auxiliary troops. The slave soldiers were the elites in both cases. In the paper by Stephen Humphreys he argues that the rise of the Mamluk armies may have been instigated by the huge numbers of slaves and migrants heading west as the Mongols expanded. It made large slave armies possible.

2. The Ayyubid empire was far less centralised than the Mamluks and was made up of regional princes with their own armies. The princes of Aleppo were recorded as using Turkmen tribes in their hinterland, while the princes of Damascus relied on Arabs from Syria and Egypt

Sources:

Gerald Hawting - Muslims, Mongols and Crusaders

David Ayalon - The Mamluk Military Society

Afaf Lutfi Al-Sayyid Marsot - A History of Egypt_ From the Arab Conquest to the Present

Stephen Humphreys - Ayyubids, Mamluks, and the Latin East in the Thirteenth Century

Stephen Humphreys - The Emergence of the Mamluk Army

Ya'acov Lev (editor) - War and Society in the Eastern Mediterranean, 7th-15th Centuries