In discussing this topic, I think it depends on who we are discussing about and not so much the subject matter itself.
If we look at the various traditions in Tibetan Buddhism, each tradition doesn't exist as exclusive isolated lineage on its own as there are shared lines of transmission. Also, all of them shared the same monastic traditions and vows from the Mulasarvastivada line.
All the schools follow a path that combines sutra and tantra, complimented with rituals and meditation. The monastic sutra topics studied are the same, for example, prajnaparamita, the middle way, correct view of voidness, pramana, and abhidharma. Some may say the texts are not the same, but even in the monasteries in the same tradition, or different colleges in the same monastery use different texts.
The different traditions practice many similar yidams, such as Chenrezig, Tara, Manjushri, Chakrasamvara and Vajrayogini. Meditation and retreats are similar, starting with preliminary practices, although there may be slight differences in structure and manner.
The lineages of teachings also mix and crossover. For example, Marpa’s transmission of The Guhyasamaja Tantra went to both Kagyus and Gelugs, and Mahamudra that is considered Kagyu are also transmitted in Sakya and Gelug schools.
Having said that, we have to reflect that what it boils down to is personal practice, hence unlike great masters, Tulkus, Geshes/Khenpos who need to transmits many teachings, initiations and practices, we should focus on one tradition as this helps to keep the component parts of the practice, the vocabulary, and the manner of expression consistent. Not many can be like Kyabje Trijang Rinpoche who can crossover the traditions, as it means that you have to master the different traditions, and yet never mixed them - when composing or teaching Gelug texts, it has to be Gelug style with its own set of concepts, vocabulary and based on the main practices and system of the tradition.
After all, our ultimate aim is to gain attainments and benefit many, I don’t think trying to climb different mountains all at the same time will bring us anywhere, and definitely we do not have time to climb so many mountains
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