Theravada has no head and no central authority. What it has instead is nikayas: monastic fraternities, each with its own ordination lineage, its own senior monks and its own internal discipline, usually recognised and regulated by the state of the country it sits in.
In Sri Lanka the Mahavihara fraternity had displaced its rivals, Abhayagiri and Jetavana, by the twelfth century; the Siyam, Amarapura and Ramanna nikayas of the eighteenth and nineteenth centuries followed, and the first of them restricted higher ordination by caste, which is a large part of why the others exist. In Thailand the Dhammayuttika order was founded as a reform by the monk who later reigned as King Mongkut from 1851 to 1868, and who thought the existing community lax in discipline and credulous in cosmology. It remains the smaller and the more prestigious of the two Thai orders.
Cutting across the fraternities is the distinction between town monasteries, which teach, chant and hold the texts, and forest monasteries, which meditate. The forest traditions of Thailand, Sri Lanka and Myanmar have periodically renewed the others, and have periodically been regarded by them as irregular.
The live dispute is over bhikkhuni ordination. The order of fully ordained nuns died out in Theravada countries centuries ago, and because ordination requires a quorum of nuns already ordained, it cannot simply be restarted from nothing. Ordinations have proceeded in Sri Lanka since the 1990s, drawing on East Asian lineages that never lapsed. Substantial parts of the Thai and Burmese hierarchies do not accept them, and women in those countries occupy positions that carry the discipline of a nun without the standing of one.