I would say that marching downhill is an awful lot harder than marching uphill. When I was in the guards, one of the most awkward jobs we had to do one year was the Tyneside Show. The showground was at the bottom of quite a steep appraoch road, and the sponsors insisted that they wanted the band to appear playing, whereas normally we would have marched down the slope and struck up once we were on the level.
There certainly are skills involved in playing on the march, particularly for brass players, when it is so easy to jar the mouthpiece against your face - much easier when I was playing saxophone, and the mouthpiece was continually between your lips. Someone mentioned previously the problems faced by a player with a long stride, and most bands make the mistake of taking too long strides when on the march. Particularly if there are women in the band, or a number of older players who are less mobile, set a shorter stride, and probably a steadier pace than you might first think, otherwise you can end up looking and feeling like penguins waddling along. If you do have hills to negotiate, you will find most bands tend to speed up when going downhill, so you need to make a conscious effort to keep the pace steady; equally, if marching along not playing there is a tendency to get faster.
Lastly, a lot of bands tend to play everything blastissimo on the march. This is counter-productive, as it is hard on the band as well as wearing for any listeners. Even if heading a march, provided the beat is still played strongly, those following will still be able to keep in step, and the effect for those at the side of the road will be much better. Rest your lips when you get the chance, make the most of the dynamic contrasts, and you will find things a lot easier. For the euphonium player finding the instrument heavy and awkward on the march, I have seen straps used, with a little ingenuity, but I have not found any solution to tired arms when playing bass trombone - it just goes with the territory!