The left/right shifts are meaningless unless you are talking specific issues. In general, I think any shift to the left on any policy is attributed to the Dems, even if neither the main party nor a Presidential candidate pushes it. Where the dems have pulled to the right, it reflects a shift in the country or a push by a few dems or liberal groups. So, the Dems aren't setting the agenda in either situation.
LBGTQIA+ (etc.) --The dems have definitely shifted left. In 2008, civil unions in lieu of gay marriage was the party line. Trans rights? Not even on the radar. The Dems shifted because it was the right position to have, and frankly America as a whole shifted. There is no shifting back, thankfully.
Black Lives Matter (police brutality) - There really isn't a policy here and I think the Dem organization as a whole has ducked the issue. But, individual dems and affiliated groups have definitely signaled a willingness to reign in cops. So, it is frequently seen as a shift with the Democrats.
DEI/race issues - Again, it's pretty light on policies. I think the Minority-Owned/Women-owned contractor policies have existed in federal contracting for a while, generally with bipartisan support. (That's a guess, since it doesn't typically get argued about.). The bigger shift left has been in corporate America, not anything driven by the Democratic Party.
Gun Control - The dems have definitely shifted to the right. Shit, I remember when there was meaningful debate about controlling hand guns. Now, shooting trophies seems to be a benefit for being a VP nom. The Democrats have crumbled on this issue. Maybe that reflects the direction the country has gone, despite the frequent murder of children.
Tax /Trade Policy - Yeah, shifted way right.
Support for Israel - unchanged, maybe a bit to the right over the last 30 years. But there are enough individual Dems and associated groups that have pushed back on Israel, so it creates the appearance of a shift in the Dem party.
Health Care - on the whole, not really changed since Hillary Clinton in the early 90s. The difference is that Obama actually got something set up. At this point, it seems like Dems are consolidating the support for Obamacare, but no wide-spread push for a better system.