So, filters can only be saved with a literal selection, or a hard-coded rule. E.g. "July", or Month Number <7.
We can't change the behavior or programmatically set the filters in Power BI natively.
We have lots of flexibility in Power Query and in DAX to create dynamic data. Since we plan on data being refreshed daily or more often, we can create a field that is defined by a formula that will indicate whether the current row has the same date as today's calendar date, for example. Assume this field is true when the row has today's date and false otherwise. Now we can set a filter to Today = True. Our filter value never changes. When our model refreshes, though, and the calendar date has changed, a different row in the date table will have the flag equal to True.
The filter is static, but the rows that meet the filter criteria change as the days pass.
Power Query has a bunch of built in date functions that help in making this sort of flag, else you can do arithmetic and conditional tests on the value of the date or associated attribute fields (e.g. year, month).
E.g. Rolling 3 months:
// Power Query
// Add custom column
// Rolling 3 month period, the 3 consecutive whole months that
// terminate at the beginning of the current month.
Rolling3MonthFlag =
let
Today = DateTime.Date( DateTime.LocalNow() )
,StartOfCurrentMonth =
#date( Date.Year( Today ), Date.Month( Today ), 1 )
,StartOf3MonthsAgo =
Date.AddMonths( StartOfCurrentMonth, -3 )
in
[Date] >= StartOf3MonthsAgo and [Date] < StartOfCurrentMonth
