Do they stack?

RAW, não.

Occasionally, your proficiency bonus might be multiplied or divided (doubled or halved, for example) before you apply it. For example, the rogue's Expertise feature doubles the proficiency bonus for certain ability checks. If a circumstance suggests that your proficiency bonus applies more than once to the same same roll, you still add it only once and multiply or divide it only once. (PHB pp.173-4, emphases mine)

Sua proficiência é dobrada apenas uma vez

No PHB p. 173:

Your proficiency bonus can’t be added to a single die roll or other number more than once. For example, if two different rules say you can add your proficiency bonus to a Wisdom saving throw, you nevertheless add the bonus only once when you make the save.

Occasionally, your proficiency bonus might be multiplied or divided (doubled or halved, for example) before you apply it. For example, the rogue’s Expertise feature doubles the proficiency bonus for certain ability checks. If a circumstance suggests that your proficiency bonus applies more than once to the same roll, you still add it only once and multiply or divide it only once.

Em geral, as coisas não se acumulam porque o PHB indica especificamente o ponto de partida:

Dragon Ancestor (PHB p.102, ênfase minha):

At 1st level, you choose one type of dragon as your ancestor. The damage type associated with each dragon is used by features you gain later.

You can speak, read, and write Draconic. Additionally, whenever you make a charisma check when interacting with dragons, your proficiency bonus is doubled if it applies to the check.

Especialização (PHB p.54, ênfase minha):

At 3rd level, choose two of your skill proficiencies. Your proficiency bonus is doubled for any ability check you make that uses either of the chosen proficiencies.

At 10th level, you can choose another two skill proficiencies to gain this benefit.

Nenhum destes modifica o seu bônus de proficiência, apenas o que é feito no teste.
Isso significa:

• dragonAncestorBonus = proficencyBonus × 2
• expertiseBonus = proficencyBonus × 2

Como ambos começam com seu bônus de proficiência, você pode escolher:

• total = roll + proficencyBonus
• total = roll + dragonAncestorBonus
• total = roll + expertiseBonus

Para ajudar a tornar as coisas mais claras, o PHB declara (p173-174):

Your proficiency bonus can't be added to a single die roll or other number more than once. For example, if two differenct rules say you can add your proficiency bonus to a wisdom saving throw, you nevertheless add the bonus only once when you make the save.

Occasionally, your proficiency bonus might be multiplied or divided (double or halved, for example) before you apply it. For example, the rogue's Expertise feature doubles the proficiency bonus for certain ability checks. If a circumstance suggests that your proficiency bonus applies more than once to the same roll, you will add it only once and multiply or divide it only once.

Para contornar essa restrição, a habilidade precisaria ser semelhante a "aumenta a proficiência em persuasão em 5" ou "adiciona 4 a todas as verificações de persuasão".

Na maioria das vezes, você não encontrará efeitos de empilhamento em D & D-5e. Se você acha que encontrou algo que está quebrado ou bom demais para ser verdade, provavelmente é.