The Solemnity of the Assumption can never be transferred to Sunday. It remains on 15 August and is a holy day of obligation in the United States, except if it falls on a Saturday or Monday. In that case, it is still a solemnity, just not one that carries an obligation to attend Mass.
Holy days have always varied from country to country and even from diocese to diocese. For example, in the United States, Epiphany and Corpus Christi are not holy days of obligation, so they are transferred to the nearest Sundays. However, in England, they are holy days of obligation, and so, remain on their actual weekdays. (Of course, Epiphany sometimes falls on a Sunday, anyway.) Similarly, the Immaculate Conception, being the patroness of the United States, is a holy day of obligation here, but not in England.
In the case of Ascension Thursday, until a decade or so ago, it was a holy day of obligation everywhere in the United States. Then permission was given by the U.S. bishops for western dioceses to transfer to feast to the following Sunday. This was followed by a proposal to do the same for all American dioceses, but was met with great opposition by some bishops, who considered this yet another rolling back of Catholic identity. (The practice of removing the obligation for some holy days when they fall on a Saturday or Monday, which had begun to be permitted a few years earlier, was cited by some bishops as an example of what they feared was becoming a trend.) The end result was a compromise, whereby bishops of each province (an archdiocese, surrounded by several dioceses) would decide whether to keep the feast as Ascension Thursday or Ascension Sunday. For example, the Province of Miami, which includes the Archdiocese of Miami and the six other dioceses in Florida, keeps Ascension Sunday. But the Province of Omaha, which includes the Archdiocese of Omaha and the Dioceses of Lincoln and Grand Island, keeps Ascension Thursday.
that most bishops who move the Feast of the Ascension to Sunday will also move the Feast of the Assumption to Sunday. And, yes, we are obliged to obey our bishops. If our bishop declares that a holy day obligation to attend Mass can be fulfilled by attending the nearest Sunday, that’s the way it is. If another bishop declares that Catholics in his diocese are obliged to attend on the holy day itself, that’s the way it is.
