I agree that it is a very difficult concept to define, and everyone's definition might differ. I would argue that any monarch who was born and raised in the realm over which they rule, speaking the local language with native fluency and identifying as a native of that land is a "native" monarch.
But in terms of defining a "native dynasty," I suppose I would go with a dynasty whose current line does not descend from a monarch who was imported (or imported him/herself by force) from abroad in conscious disregard of an existing or potential native born claimant. This would be regardless of whether that imported/usurping/conquering monarch had ancestors from that land.
Thus the current European dynasties, whose current monarchs are native born all, would be disqualified:
Bernadotte - Karl XIV Johan imported from France
Glucksburg of Denmark - Christian IX imported from Germany
Glucksburg of Norway - Haakon VII invited and confirmed by referendum
Grimaldi - Rainier I conquered from Genoa
Liechtenstein - first prince to live in principality was Franz Josef II
Orange of the Netherlands - Of foreign origin but native at the time the monarchy was created
Orange-Nassau - Adolphe came from Germany due to Salic Law
Windsor - George I imported from Hanover
it's a tricky thing as the House of Orange-Nassau had been in the Netherlands since the 15th century and only became the reigning dynasty in 1814 so if a family has not become part of the country in 4 centuries what does that say about people in the US, Canada, Latin America, Australia etc?
Most reigning dynasties have links with native dynasties before them. The Belgian Coburgs through their Habsburg ancestry descent from the dukes of Burgundy who in turn were descending from earlier houses reigning in Flanders and Brabant.
The House of Bourbon-Parma descents from Elisabeth Farnese so has it's links to the previous dynasty.
The Windsors do descent from the Stuarts, Tudors and Plantegenets before them.
The Spanish Borbons have female line links to previous Iberian dynasties.
Though Carl XVI Gustav is a Bernadotte he is also the most senior member of the Holstein-Gottorp-Wasa line through Victoria of Baden and descents from Gustav IV as well as his adoptive first cousin Carl XIV Johan.
The Grimaldi's did not start out as a family in Monaco, i believe they came from Genua, but have been there for centuries.
I suppose Liechtenstein might fit your description as the dynasty gave it's name to the country.
Depends on what you call native.
The house of Orange was a branch of the germanic house of Nassau, whose site is in Rhineland-Palatinate, quite far from the NL.
The Ottonian line of the Nassaus established in the dutch region but they were not "native".
I won't enter on the discussion that both lines of the Nassaus are extinct in the male line, the Ottonians with Wilhelmina, the Walramians with Charlotte of Luxembourg.
The Grimaldis of Monaco are native from neighbouring Genoa.
As to the Orange, they are also extinct in the male line, when Louise-Hypolite Grimaldi married Jacques de Goyon de Matignon, who assumed the name and arms of his wife's family.
Of the European monarchies today, are the Houses of Orange and Grimaldi the only dynasties whom we might consider to be "native" in the countries over which they reign? Those countries whose dynasties are represented solely by a "crown prince" would Montenegro, Albania and Serbia be the only ones in this category?