Though I'm no Miss Manners, I'd say that if you're on a tight budget that extravagant "nice" gifts do not have to be. Tis true that the thought is what counts....
Four year old's birthday party... does she have long hair? Go to the dollar store and load up on cute pony tail holders, a new brush, some children's fingernail polish.... or a summertime fun bucket, a little sand pail, shovel, bottle of bubbles, inflatable ball and some sidewalk chalk...that would be an inexpensive but cute way to present it too..
One thing you can do when giving gifts to children -- nix the cards. They don't really care anyhow... just fold a little piece of cardstock in half, write to/from inside and attach it with some curly ribbon.... voila...and you saved yourself $3 on a card that will be thrown away in seconds.
If you have a very good family/friend birthday...go to the grocery store and buy her a little bouquet of flowers and bring that. Those are generally..what... 3 small packs for $10.
Parent's anniversary... yeah, I'd think that would should be something thought out... why not a gift card to their favorite restaurant. My in laws love that.
When does moving = presents? I've never heard of that and I've moved A LOT.... I'd say one very thoughtful gift if you felt you MUST give one is to purchase a set of note cards and a small book of stamps and a colorful pen (not an expensive one, just one for some color) and just put in a little gift bag and put something like "hope this helps us keep in touch"....... the children... depends on how old they are but I'd say a pad of white paper and some markers and envelopes that are PREADDRESSED to your children so that they can keep in touch... but I wouldnt' think you owe kids anything unless you're doing something like a moving sanity kit with a bag of snacks for the road trip....
Just because you feel compelled TO give gifts does not mean you HAVE to... esp the movers...